#OEIS

One day, one decomposition
A013916: Numbers k such that the sum of the first k primes is prime

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A013916.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A013916.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #sum #first #primes #primenumbers #graph

Decomposition into weight × level + jump of A013916 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A013916 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

2 days ago

One day, one decomposition
A011775: Numbers k such that k divides phi(k) * sigma(k)

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A011775.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A011775.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #divides #phi #sigma #graph

Decomposition into weight × level + jump of A011775 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A011775 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

3 days ago

One day, one decomposition
A011540: Numbers that contain a 0

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A011540.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A011540.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #contain #digit #graph

Decomposition into weight × level + jump of A011540 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A011540 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

4 days ago

One day, one decomposition
A011539: "9ish numbers": decimal representation contains at least one nine

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A011539.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A011539.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #contain #digit #graph

Decomposition into weight × level + jump of A011539 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A011539 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

5 days ago

One day, one decomposition
A011533: Numbers that contain a 3

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A011533.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A011533.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #contain #digit #graph

Decomposition into weight × level + jump of A011533 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A011533 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

6 days ago

One day, one decomposition
A011532: Numbers that contain a 2

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A011532.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A011532.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #contain #digit #graph

Decomposition into weight × level + jump of A011532 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A011532 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

1 week ago

One day, one decomposition
A011531: Numbers that contain a 1

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A011531.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A011531.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #contain #digit #graph

Decomposition into weight × level + jump of A011531 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A011531 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

1 week ago

One day, one decomposition
A011257: Geometric mean of phi(n) and sigma(n) is an integer

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A011257.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A011257.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #phi #sigma #graph

Decomposition into weight × level + jump of A011257 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A011257 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

1 week ago

One day, one decomposition
A010784: Numbers with distinct decimal digits

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A010784.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A010784.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #distinct #decimal #digits #graph

Decomposition into weight × level + jump of A010784 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A010784 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

1 week ago

50 sequences decomposed into weight × level + jump in one GIF

1000 sequences decomposed with 3D graphs (threejs webGL), 2D graphs - CSV, dump, img to download
⬇️⬇️
https://decompwlj.com

#decompwlj ➡️ https://oeis.org/wiki/Decomposition_into_weight_*_level_%2B_jump

#Math #Mathematics #OEIS #sequences #primes #PrimeNumbers #classification #arithmetic #NumberTheory #JavaScript #GIF #graphs

50 sequences decomposed into weight × level + jump in one GIF with A-numbers (log(weight), log(level))

Rémi Eismann

1 week ago

One day, one decomposition
A010064: Base 4 self or Colombian numbers (not of form n + sum of base 4 digits of n)

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A010064.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A010064.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #numbers #3D #sum #base #digits #graph

Decomposition into weight × level + jump of base 4 self or Colombian numbers in 2D (log(weight), log(level))

Decomposition into weight × level + jump of base 4 self or Colombian numbers in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

2 weeks ago

My first, favorite and most important sequence, the weights of prime numbers
⬇️
http://oeis.org/A117078

We see prime numbers classified by level and by weight on the graph.

This sequence has led to the decomposition into weight × level + jump ➡️ https://oeis.org/wiki/Decomposition_into_weight_*_level_%2B_jump

#decompwlj #math #sequences #OEIS #PrimeNumbers #classification #mathematics #primes #numbers #graph #decomposition #Numbertheory

Screen shot of A117078 on the OEIS: a(n) is the smallest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists.

Graphs on the OEIS of A117078 (the weights of prime numbers) pin plot and logarithmic scatterplot

Rémi Eismann

2 weeks ago

One day, one decomposition
A010061: Binary self or Colombian numbers: numbers that cannot be expressed as the sum of distinct terms of the form 2^k+1 (k>=0), or equivalently, numbers not of form m + sum of binary digits of m

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/Binary_self_numbers.html
2D #graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/Binary_self_numbers.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #sum #binary #digits #graph

Decomposition into weight × level + jump of Binary self or Colombian numbers in 2D (log(weight), log(level))

Decomposition into weight × level + jump of Binary self or Colombian numbers in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

2 weeks ago

One day, one decomposition
A009177: #Numbers that are the hypotenuses of more than one Pythagorean triangle

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A009177.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A009177.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #hypotenuses #Pythagorean #triangle #graph

Decomposition into weight × level + jump of A009177 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A009177 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

2 weeks ago

One day, one decomposition
A009112: Areas of Pythagorean triangles: numbers which can be the area of a right-angled triangle with integer sides

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A009112.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A009112.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #area #Pythagorean #triangles #graph

Decomposition into weight × level + jump of A009112 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A009112 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

2 weeks ago

150 sequences decomposed into weight × level + jump in one GIF

1000 sequences decomposed with 3D graphs (threejs webGL), 2D graphs - CSV, dump, img to download
⬇️⬇️
https://decompwlj.com

#decompwlj ➡️ https://oeis.org/wiki/Decomposition_into_weight_*_level_%2B_jump

#Math #Mathematics #OEIS #sequences #primes #PrimeNumbers #classification #arithmetic #NumberTheory #JavaScript #GIF #graphs

150 sequences decomposed into weight × level + jump in one GIF with A-numbers (log(weight), log(level))

Rémi Eismann

2 weeks ago

One day, one decomposition
A008917: Numbers that are the sum of 3 positive cubes in more than one way

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A008917.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A008917.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #sum #cubes #graph

Decomposition into weight × level + jump of A008917 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A008917 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

2 weeks ago

One day, one decomposition
A008864: a(n) = prime(n) + 1

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A008864.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A008864.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #prime #primenumbers #graph

Decomposition into weight × level + jump of A008864 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A008864 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

2 weeks ago

One day, one decomposition
A008851: Congruent to 0 or 1 mod 5

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A008851.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A008851.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #congruent #modulo #graph

Decomposition into weight × level + jump of A008851 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A008851 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

2 weeks ago

One day, one decomposition
A008846: Hypotenuses of primitive Pythagorean triangles

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A008846.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A008846.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #triangles #graph

Decomposition into weight × level + jump of A008846 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A008846 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

3 weeks ago

One day, one decomposition
A008810: a(n) = ceiling(n^2/3)

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A008810.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A008810.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #ceiling #graph

Decomposition into weight × level + jump of A008810 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A008810 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

3 weeks ago

My first, favorite and most important sequence, the weights of prime numbers
⬇️
http://oeis.org/A117078

We see prime numbers classified by level and by weight on the graph.

This sequence has led to the decomposition into weight × level + jump ➡️ https://oeis.org/wiki/Decomposition_into_weight_*_level_%2B_jump

#decompwlj #math #sequences #OEIS #PrimeNumbers #classification #mathematics #primes #numbers #graph #decomposition #Numbertheory

Rémi Eismann

3 weeks ago

One day, one decomposition
A008366: Smallest prime factor is >= 17

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A008366.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A008366.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #rough #numbers #divisible #graph

Decomposition into weight × level + jump of A008366 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A008366 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

3 weeks ago

One day, one decomposition
A008365: Smallest prime factor is >= 13

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A008365.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A008365.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #rough #numbers #divisible #prime #factor #graph

Decomposition into weight × level + jump of A008365 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A008365 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

3 weeks ago

One day, one decomposition
A008364: 11-rough numbers: not divisible by 2, 3, 5 or 7

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/11-rough_numbers.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/11-rough_numbers.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #rough #numbers #divisible #graph

Decomposition into weight × level + jump of 11-rough numbers in 2D (log(weight), log(level))

Decomposition into weight × level + jump of 11-rough numbers in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

3 weeks ago

One day, one decomposition
A007957: Numbers that contain an odd digit

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A007957.html
2D #graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A007957.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #contain #odd #digit #graph

Decomposition into weight × level + jump of A007957 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A007957 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Peter Luschny

3 weeks ago

#OEIS A352178

On computing sets of integers with maximum number of pairs summing to powers of 2.

Max A. Alekseyev https://arxiv.org/abs/2303.02872

Originated in April 2021 with Dan Ullman's and Stan Wagon's "Problem of the Week 1321".

It was noted that the problem has a natural interpretation as finding a maximal graph of order n, where the vertices are labeled with pairwise distinct integers and the sum of the endpoint labels for each edge is a power of 2.

Numberphile Youtube https://youtu.be/IPoh5C9CcI8

Rémi Eismann

3 weeks ago

One day, one decomposition
A007932: Numbers that contain only 1's, 2's and 3's

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A007932.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A007932.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #contain #only #digits #graph

Decomposition into weight × level + jump of A007932 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A007932 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

3 weeks ago

One day, one decomposition
A007921: Numbers that are not the difference of two primes

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A007921.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A007921.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #difference #primes #primenumbers #graph

Decomposition into weight × level + jump of A007921 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A007921 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

4 weeks ago

One day, one decomposition
A007811: Numbers n for which 10n+1, 10n+3, 10n+7 and 10n+9 are primes

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A007811.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A007811.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #primes #primenumbers #form #graph

Decomposition into weight × level + jump of A007811 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A007811 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

4 weeks ago

One day, one decomposition
A007770: Happy numbers: numbers whose trajectory under iteration of sum of squares of digits map (see A003132) includes 1

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/Happy_numbers.html
2D #graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/Happy_numbers.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #trajectory #iteration #sum #squares #digits #map #graph

Decomposition into weight × level + jump of happy numbers in 2D (log(weight), log(level))

Decomposition into weight × level + jump of happy numbers in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

4 weeks ago

50 sequences decomposed into weight × level + jump in one GIF

1000 sequences decomposed with 3D graphs (threejs webGL), 2D graphs - CSV, dump, img to download
⬇️⬇️
https://decompwlj.com

#decompwlj ➡️ https://oeis.org/wiki/Decomposition_into_weight_*_level_%2B_jump

#Math #Mathematics #OEIS #sequences #primes #PrimeNumbers #classification #arithmetic #NumberTheory #JavaScript #GIF #graphs

Rémi Eismann

1 month ago

One day, one decomposition
A007703: Regular primes

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/Regular_primes.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/Regular_primes.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #Bernoulli #numbers #divisible #primes #primenumbers #graph

Decomposition into weight × level + jump of regular primes in 2D (log(weight), log(level))

Decomposition into weight × level + jump of regular primes in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

1 month ago

My first, favorite and most important sequence, the weights of prime numbers
⬇️
http://oeis.org/A117078

We see prime numbers classified by level and by weight on the graph.

This sequence has led to the decomposition into weight × level + jump ➡️ https://oeis.org/wiki/Decomposition_into_weight_*_level_%2B_jump

#decompwlj #math #sequences #OEIS #PrimeNumbers #classification #mathematics #primes #numbers #graph #decomposition

Rémi Eismann

1 month ago

One day, one decomposition
A007700: Numbers n such that n, 2n+1, and 4n+3 all prime

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A007700.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A007700.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #primes #primenumbers #graph

Decomposition into weight × level + jump of A007700 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A007700 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

1 month ago

One day, one decomposition
A007692: Numbers that are the sum of 2 nonzero squares in 2 or more ways

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A007692.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A007692.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #sum #squares #ways #graph

Decomposition into weight × level + jump of A007692 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A007692 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

1 month ago

One day, one decomposition
A007618: a(n) = a(n-1) + sum of digits of a(n-1), a(1) = 5

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A007618.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A007618.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #sum #digits #graph

Decomposition into weight × level + jump of A007618 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A007618 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Peter Luschny

1 month ago

#OEIS A360923

The Integer Lunar Lander

A position in the game of Integer Lunar Lander consists of an ordered pair (i, j) of integers. There are always 3 legal moves, to (i+1, i+j+1), (i, i+j), and (i-1, i-j-1). The object of the game is to reach (0, 0) in the minimum possible number of moves T(i, j).

The ordered pair may be interpreted as the upward velocity and altitude of a vehicle; the object is to land the vehicle at zero velocity.

Rémy Sigrist's scatterplot:
https://oeis.org/A360923/a360923_1.png

Rémi Eismann

1 month ago

One day, one decomposition
A007606: Take 1, skip 2, take 3, etc

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A007606.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A007606.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #take #skip #graph

Decomposition into weight × level + jump of A007606 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A007606 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

1 month ago

One day, one decomposition
A007532: Powerful numbers (2): a sum of positive powers of its digits

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A007532.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A007532.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #powerful #numbers #sum #powers #digits #graph

Decomposition into weight × level + jump of A007532 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A007532 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Peter Luschny

1 month ago

#OEIS

Do you want to sell the OEIS?

No problem. The OEIS has changed its license to CC-BY-SA-4.0.

The license requires attribution when parts of the OEIS are reproduced.

The new license allows commercial use provided that the result is distributed under the same conditions ("share-alike").

https://creativecommons.org/licenses/by-sa/4.0/

Peter Luschny

1 month ago

#OEIS

Do you want to clone the OEIS?
No problem!

https://github.com/oeis/oeisdata

Christian Lawson-Perfect

1 month ago

Here's an integer sequence I came up with that I think could be a decent #LullabySequence.
Not in the #OEIS yet - I've got a queue of three to submit now, I think!

2,4,1,5,9,13,17,12,7,2,8,...

As usual: your best guesses for the next terms and the underlying rule, under a content warning, please!

Rémi Eismann

1 month ago

One day, one decomposition
A007066: a(n) = 1 + ceiling((n-1)*phi^2), phi = (1+sqrt(5))/2

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A007066.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A007066.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #ceiling #phi #sqrt #graph

Decomposition into weight × level + jump of A007066 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A007066 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

1 month ago

50 sequences decomposed into weight × level + jump in one GIF

1000 sequences decomposed with 3D graphs (threejs webGL), 2D graphs - CSV, dump, img to download
⬇️⬇️
https://decompwlj.com

#decompwlj ➡️ https://oeis.org/wiki/Decomposition_into_weight_*_level_%2B_jump

#Math #Mathematics #OEIS #sequences #primes #PrimeNumbers #classification #arithmetic #NumberTheory #JavaScript #GIF #graphs

Rémi Eismann

2 months ago

My first, favorite and most important sequence, the weights of prime numbers
⬇️
http://oeis.org/A117078

We see prime numbers classified by level and by weight on the graph.

This sequence has led to the decomposition into weight × level + jump ➡️ https://oeis.org/wiki/Decomposition_into_weight_*_level_%2B_jump

#decompwlj #math #sequences #OEIS #PrimeNumbers #classification #mathematics #primes #numbers #graph #decomposition

Rémi Eismann

2 months ago

One day, one decomposition
A006532: Numbers whose sum of divisors is a square

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A006532.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A006532.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #sum #divisors #square #graph

Decomposition into weight × level + jump of A006532 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A006532 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Spektrum (inoffiziell)

2 months ago

Dass Primzahlen und Zweierpotenzen viele Menschen faszinieren, ist nicht überraschend. Tatsächlich spalten sich alle Zahlen in zwei Lager auf: interessante und langweilige. Warum?
Die fabelhafte Welt der Mathematik: Auf der Suche nach der langweiligsten Zahl der Welt
#Zahlentheorie #FabelhafteMathematik #Zahlen #Sloane #OEIS #Primzahlen #Mathematik

Peter Luschny

2 months ago

#OEIS
@johncarlosbaez @Lisanne

https://oeis.org/A359146
(is still under construction)

https://www.nytimes.com/2023/02/07/science/puzzles-rectangles-mathematics.html

Peter Luschny

2 months ago

#OEIS

OEIS Foundation Seeks to Raise $3M Endowment to Fund Full-Time Managing Editor

https://oeisf.org/fund-2023-02-02.pdf

Peter Luschny

2 months ago

#OEIS

The OEIS style sheet says:

"Avoid vanity: do not name a sequence after yourself (or your family members, your friends, etc.)."

What bad luck! Now that I've found primes I'd like to call 'Peter Primes' (if only for the sake of alliteration.)

So I called them 'primitive prime powers'. They start:

5, 17, 37, 97, 149, ...

But maybe that's for the best, because I have no proof that they're really prime numbers.

If you can prove it, secure eternal glory and enter it in A360204!

John May

2 months ago

A weekend of numbers: Saturday I was number 46 in the LA Tourist race, Sunday I was number 32 in the Coaster Brake Challenge, Monday I was juror number 62 in LA county superior court, so I went to #OEIS to see what number I am today. 41 apparently

Peter Luschny

2 months ago

#OEIS #A359762 #A000085

(4/4)

The generalized telephone numbers.

Urszula Bednarz and Małgorzata Wołowiec-Musiał, On a new generalization of telephone numbers.

(Python)

from math import factorial, comb
def oddfactorial(n: int) -> int:
return factorial(2 * n) // (2**n * factorial(n))

def T(n: int, k: int) -> int:
return sum(comb(n, 2 * j) * oddfactorial(j) * k**j for j in range(n + 1))

for n in range(10): print([T(n, k) for k in range(8)])

k = 1 gives the standard telefon numbers.

Peter Luschny

2 months ago

#OEIS #A359762

(3/4)

In graph theory, the n-th telephone number is the total number of matchings of a complete graph K_n.

Assuming a telephone system with k possibilities of connections and n subscribers leads to a network that can be modeled by a complete multigraph K(n, k). The total number of connection patterns in such a network is given by A359762.

https://en.wikipedia.org/wiki/Complete_graph

Peter Luschny

2 months ago

#OEIS #A000085

(2/4)

There are a variety of formulas for the calculation of the telephone numbers.

In the OEIS you can find three different Sage implementations, one more beautiful than the other.

def A(n): return (i/sqrt(2))^n*hermite(n, -i/sqrt(2))

def B(n): return hypergeometric([-n/2, (1-n)/2], [], 2).simplify()

def C(n): return sum(factorial(n) // (factorial(n-2*k) * 2**k * factorial(k)) for k in range(1+n//2))

for f in [A, B, C]: print([f(n) for n in range(20)])

Peter Luschny

2 months ago

#OEIS #A000085

(1/4)

The number of involutions of permutations on n letters, is also known as 'telephone numbers'. According to Bednarz et al. the telephone number interpretation "is due to John Riordan, who noticed that T(n) is the number of connection patterns in a telephone system with n subscribers."

Plot from Wikipedia, Author David Eppstein.

https://en.wikipedia.org/wiki/Telephone_number_(mathematics)

The complete graph K4 has ten matchings, corresponding to the value T(4) = 10 of the fourth telephone number.

Christian Lawson-Perfect

2 months ago

@kimreece there's an entry in the #OEIS giving the end of this process for each starting number: http://oeis.org/A239448

Peter Luschny

2 months ago

#OEIS #A358073

Another mysterious scatter plot.
Author: John Tyler Rascoe

Peter Luschny

3 months ago

@fredrikj
#OEIS #A000045

Impressive!

"For people who care about computing the billionth Fibonacci number ..."
Has the existence already been asserted? :fibonacci: 😂

And here is the challenge for the common man (Julia speaking):

function fibrec(n::Int)
n == 0 && return (BigInt(0), BigInt(1))
a, b = fibrec(div(n, 2))
c = a * (b * 2 - a)
d = a * a + b * b
iseven(n) ? (c, d) : (d, c + d)
end

fibonacci(n::Int) = fibrec(n)[1]
println([fibonacci(n) for n in 0:40])

Peter Luschny

3 months ago

#OEIS #A359703

A new sequence of Don Knuth.

Number of fillomino dissections of a 2 X n rectangle.

A fillomino dissection of a rectangle is a tiling by polyominoes in which no two polyominoes of the same size are adjacent.

https://oeis.org/A359652

https://en.wikipedia.org/wiki/Fillomino

Peter Luschny

3 months ago

#OEIS #SLOANE

What a nice coincidence! Neil Sloane uploaded a paper to the arXiv today. The abstract says:

"Someone coming across the sequence 1, 2, 4, 9, 21, 51, 127, . would have had no way (before 1973) of discovering that it had been studied since 1870 (today these are called the Motzkin numbers, and form entry A001006 in the database)."

And I published an entry today on the Motzkin triangle, a refinement of the Motzkin numbers.

https://oeis.org/A359364

https://arxiv.org/abs/2301.03149

Toby Walsh

3 months ago

The history of a fantastic resource #OEIS https://arxiv.org/pdf/2301.03149.pdf

LaPingvino 🟙 :ir:

3 months ago

@hrefna I implemented #LocationLabels using #UlamSpirals, for which I used the #OEIS to find an algorithm to iterate over the spiral (implemented at whenwhere.cf)

Christian Lawson-Perfect

3 months ago

I've come up with an integer sequence that doesn't seem to be in the #OEIS yet. Here are the first few terms:

1,2,2,4,3,6,7,8,4,10

Can you work out the rule? I can reveal more terms on request, under a content warning. If you have an idea, please CW it so other people can choose not to see hints

Rémi Eismann

3 months ago

One day, one decomposition
A004780: Binary expansion contains 2 adjacent 1's

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A004780.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A004780.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #number #binary #expansion #contain #adjacent #graph

Decomposition into weight × level + jump of A004780 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A004780 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Rémi Eismann

3 months ago

One day, one decomposition
A004744: Numbers n such that binary expansion does not contain 011

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A004744.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A004744.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #binary #expansion #contain #graph

Decomposition into weight × level + jump of A004744 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A004744 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Christian Lawson-Perfect

3 months ago

Came up with another #LullabySequence while getting the toddler to sleep:

1,2,4,3,6,5,10,9,8,7,14,...

This one's already in the #OEIS, so you can spoil yourself if you can't see the pattern

Peter Luschny

3 months ago

#OEIS A359413

Knuth considered sorting permutations by passing them through a stack. Variations include allowing multiple passes through the stack and using different data structures.

A. Claesson & B. Á. Guðmundsson use pop-stacks in their paper
https://arxiv.org/abs/1710.04978

They show that the generating functions for perms. sortable by k passes are rational for any k. You can find the first six g.f.s on GitHub:
https://github.com/SuprDewd/popstacks

There is also a nice Python program in the OEIS entry.

Peter Luschny

3 months ago

#OEIS A001181 and A359363.

A Baxter permutation (BP) on {1,2,...,n} is a permutation p = (p1 p2 ... pn) for which there are no quadruples of indices
i < j < j + 1 < k such that
pj < pk < pi < p(j+1) or p(j+1) < pi < pk < pj.

In other words, BPs avoid certain permutation patterns: 2-41-3 and 3-14-2.

There are 1-1 correspondences between BPs and interesting combinatorial objects like floorplan partitions.

This year Don Knuth's Christmas Lecture was about BPs.
https://www.youtube.com/watch?v=zg6YRqT4Duo

John May

3 months ago

I got a nudge to donate to the #OEIS before the year-end, and you know because, I knew I couldn't just donate a round dollar amount, so I donated $0111.21 in honor of A000001. Here's your nudge. Find a sequence and move the decimal place to somewhere you feel comfortable. https://oeisf.org/donate/

Christian Lawson-Perfect

3 months ago

I came up with an integer sequence that isn't in the #OEIS, but it's silly:

1,2,2,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,8, …

Can you give the next 57 terms?

Rémi Eismann

3 months ago

One day, one decomposition
A004214: Positive numbers that are not the sum of three nonzero squares

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A004214.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A004214.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #sum #three #squares #graph

Decomposition into weight × level + jump of A004214 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A004214 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

3 months ago

I use #oeis a lot when working through problem sets for #SICP. I feel like it's not cheating though because that last part of coming up with the general formula for describing an integer sequence seems orthogonal to actually getting the sequence itself? I mean I'm doing the experiments and getting the numbers myself?

idk maybe i'm wrong ?

Peter Luschny

3 months ago

Mark the points of the Farey series F_n on a strip of paper and wrap it around a circle of circumference 1 so the endpoints 0 and 1 coincide; draw a chord between every pair of the Farey points; a(n) is the number of vertices in the resulting graph.

#OEIS https://oeis.org/A359116

Rémi Eismann

3 months ago

My first, favorite and most important sequence, the weights of prime numbers
⬇️
http://oeis.org/A117078

We see prime numbers classified by level and by weight on the graph.

This sequence has led to the decomposition into weight × level + jump ➡️ https://oeis.org/wiki/Decomposition_into_weight_*_level_%2B_jump

#decompwlj #math #sequences #OEIS #PrimeNumbers #classification #mathematics #primes #numbers #graph #decomposition

Screen shot of A1170078 on the OEIS: a(n) is the smallest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists.

Rémi Eismann

3 months ago

One day, one decomposition
A003628: Primes congruent to {5, 7} mod 8

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A003628.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A003628.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #primes #PrimeNumbers #congruent #graph

Decomposition into weight × level + jump of A003628 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A003628 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Peter Luschny

4 months ago

A Beatty sequence is the sequence of integers obtained by taking the floor of the positive multiples of a positive irrational number.

A Cloitre walk is a certain algorithm to visualize a binary sequence.

Sela Fried, in 'The Beauty of Beatty sequences', gives a gallery of 100 of his favorite Cloitre walks.

Plus Python code to generate Cloitre walks of the Beatty sequences B(sqrt(n)).

Download from https://oeis.org/A083035

#BeattySequence #CloitreWalk
#symmetry #OEIS

Jon Awbrey

4 months ago

@christianp @jasonc @andrejbauer

#OEIS #PrimeTime #RiffsAndRotes
#DoublyRecursiveFactorization ☟

• https://oeis.org/wiki/Riffs_and_Rotes

Rémi Eismann

4 months ago

My first, favorite and most important sequence, the weights of prime numbers
⬇️
http://oeis.org/A117078

We see prime numbers classified by level and by weight on the graph.

This sequence has led to the decomposition into weight × level + jump ➡️ https://oeis.org/wiki/Decomposition_into_weight_*_level_%2B_jump

#decompwlj #math #sequences #OEIS #PrimeNumbers #classification #mathematics #primes #numbers #graph #decomposition

4 months ago

almost done with chapter 1. I have to admit for the exercises I often use #oeis often as an assistant - just because I can't be damned to figure out what the sequences are.

Rémi Eismann

4 months ago

50 sequences decomposed into weight × level + jump in one GIF

1000 sequences decomposed with 3D graph (#threejs #webGL), 2D graphs - CSV, dump, img to download
⬇️⬇️
https://decompwlj.com

#decompwlj #Math #Mathematics #OEIS #sequences #primes #PrimeNumbers #classification #arithmetic #NumberTheory #JavaScript #GIF #graphs

Bill Tozier

4 months ago

Mentioned in passing in an old blog post I am finally reading (Playful Math Education, Blog Carnival #157 at http://mathmamawrites.blogspot.com/2022/07/math-teachers-at-play-aka-playful-math.html) is the observation that 157 is the 37th prime, and 37 is also prime.

Now 37 is the 12th prime, and 12 is not prime. That makes me sad but intrigued.

What is the longest chain of prime numbers of this sort? That is "N1 is the N2th prime, and N2 is the N3rd prime, and N3 is the N4th prime…" and so on?

#NumberTheory #RecreationalMath #OEIS ?

Rémi Eismann

4 months ago

My first, favorite and most important sequence, the weights of prime numbers
⬇️
http://oeis.org/A117078

We see prime numbers classified by level and by weight on the graph.

This sequence has led to the decomposition into weight × level + jump ➡️ https://oeis.org/wiki/Decomposition_into_weight_*_level_%2B_jump

#decompwlj #math #sequences #OEIS #PrimeNumbers #classification #mathematics #primes #numbers #graph #decomposition

Ryan McShane, Ph.D.

4 months ago

I spent the day yesterday writing my first ever R package!!! :blobcatheartbongo: It's installable from GitHub (with a PAT) at the moment, but is missing a lot of functions. :blobcat_glitch:

This package is in support of a paper I am writing which should result in more integer sequences being added to the OEIS (A357257, A357248, and A357266 are already approved), related to my work on intransitivity in paired comparisons :fediverse:

#RStats #oeis @rstats

A document() call showing a dozen or so functions being documented (.Rd files being generated) as well as the help file preview of a function I have called split.atom.

John May

5 months ago

I still haven't solved the mystery of why I have a worksheet working on this problem. But I did find visit the base 10 primes in the #OEIS A085823 and it links this paper posted on the arχiv https://arxiv.org/abs/1912.08598 and that paper links a couple math stackexchange questions, which might explain it a little. But this answer explains the stripes https://math.stackexchange.com/a/3247061/416418 "the length [a substring prime base b] is at most 2𝑝−1 where 𝑝 is the smallest prime that does not divide b"

John May

5 months ago

After a couple tries, I found this sequence of #primenumber counts on #OEIS (you have to include the single digit primes) http://oeis.org/A309572
The OEIS has table of the first 2500 entries and a log scale plot which I recreated here in Maple. The submitter, Rémy Sigrist, identified the pattern based on residues mod 6. Sadly there is no additional work linked.

I've added more colors for b ≡ 0 (mod 30 and 210) and it DOES look like the size depends on the distinct number of primes factors

Logarithmic scatter plot of the counts of number of "substring primes" from OEIS A309572, primes appear in distinct groups depending on their number of distinct prime factors with more factors correlating to larger counts.

Christian Lawson-Perfect

6 months ago

I have a new #OEIS sequence: " a(n) is the smallest k such that k! has at least n consecutive zeros immediately after the leading digit in base 10."
http://oeis.org/A356739

Christian Lawson-Perfect

7 months ago

In case you were wondering, 153 is the smallest integer whose factorial has two zeros after the leading digit. It fits in a toot:

153! = 200634390509568239477828874698911718566246149616161171934231099284840946025238092339613294062603588435530393145048663047173051913507711632216305667129554900620296603188543122491838966881134795135997316305640071571629943041039657861120000000000000000000000000000000000000

This has generated a *superb* integer sequence which isn't yet in the OEIS

#math #oeis

kdkeller

2 years ago

@xpil Hello xpil,

thanks for your interesting problem.
I thougt about the knight moves in 3D and made
a model with #GeoGebra: https://www.geogebra.org/3d/awb3hzph

There are 48 moves that lead to 24 fields.

The sequences/functions you mentioned are on #OEIS
http://oeis.org/A024023 and http://oeis.org/A033996

Greetings!