cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A343491 Number of representations of n! as a sum of 3 tetrahedral numbers (A000292).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 1, 3, 5, 2, 3, 6, 5, 8, 8, 7, 2, 7, 8, 3, 11, 2, 2
Offset: 1

Views

Author

Altug Alkan, Apr 17 2021

Keywords

Comments

Conjecture I: There are infinitely many n such that a(n) >= 1.
Conjecture II: Natural density of numbers n such that a(n) >= 1 is 1.
Conjecture III: Numbers n such that a(n) = 0 is a finite sequence.
Conjecture IV: a(n) >= 1 for all n.
See Links section for some solutions.

Examples

			a(4) = 2 because 4! = 0 + 4 + 20 = 4 + 10 + 10.
a(24) = 2 because 24! = f(11393630) + f(118661018) + f(127041924) = f(81298034) + f(61098204) + f(143537134) where f = A000292.

Links

Crossrefs

Cf. A000142, A000292, A102796, A104246, A102799, A267414, A308852, A341794.

Programs

  • Mathematica
    Table[Length[Solve[{i*(i + 1)*(i + 2) + j*(j + 1)*(j + 2) + k*(k + 1)*(k + 2) == 6*n!, i >= 0, j >= 0, k >= 0, i <= j, j <= k, k < (6*n!)^(1/3)}, Integers]], {n, 1, 10}] (* Vaclav Kotesovec, Apr 19 2021 *)

A102804 Let f(n) be the minimal number of distinct nonzero tetrahedral numbers that add to n (or -1 if n is not a sum of distinct tetrahedral numbers); sequence gives numbers n for which f(n) = 5.

Original entry on oeis.org

106, 116, 122, 153, 171, 174, 191, 207, 216, 219, 229, 236, 238, 246, 252, 267, 271, 274, 283, 298, 319, 329, 336, 338, 355, 357, 367, 382, 383, 393, 401, 408, 414, 432, 433, 435, 437, 438, 447, 454, 467, 474, 477, 492, 499, 513, 518, 528
Offset: 1

Views

Author

Jud McCranie, Feb 26 2005

Keywords

Crossrefs

Cf. A000292, A104246, A102795, etc.

A102805 Let f(n) be the minimal number of distinct nonzero tetrahedral numbers that add to n (or -1 if n is not a sum of distinct tetrahedral numbers); sequence gives numbers n for which f(n) = 6.

Original entry on oeis.org

126, 392, 402, 418, 439, 457, 464, 502, 538, 577, 587, 602, 612, 638, 657, 722, 793, 812, 822, 838, 863, 1007, 1062, 1198, 1408, 1423
Offset: 1

Views

Author

Jud McCranie, Feb 26 2005

Keywords

Comments

It appears that there are just two numbers n for which f(n) = 7, namely 412 and 622 and none with f(n) > 7.

Crossrefs

Cf. A000292, A104246, A102795, etc.

A356037 Conjecturally, a(n) is the smallest number m such that every natural number is a sum of at most m n-simplex numbers.

Original entry on oeis.org

1, 3, 5, 8, 10, 13, 15, 15, 19, 24
Offset: 1

Views

Author

Mohammed Yaseen, Jul 24 2022

Keywords

Comments

n-simplex numbers are {binomial(k,n); k>=n}.
This problem is the simplex number analog of Waring's problem.
a(2) = 3 was proposed by Fermat and proved by Gauss, see A061336.
Pollock conjectures that a(3) = 5. Salzer and Levine prove this for numbers up to 452479659. See A104246 and A000797.
Kim gives a(4)=8, a(5)=10, a(6)=13 and a(7)=15 (not proved).

Examples

			2-simplex numbers are {binomial(k,2); k>=2} = {1,3,6,10,...}, the triangular numbers. 3 is the smallest number m such that every natural number is a sum of at most m triangular numbers. So a(2)=3.
3-simplex numbers are {binomial(k,3); k>=3} = {1,4,10,20,...}, the tetrahedral numbers. 5 is presumed to be the smallest number m such that every natural number is a sum of at most m tetrahedral numbers. So a(3)=5.

Links

Crossrefs

Cf. A002804, A079611.
Minimal number of x-simplex numbers whose sum equals n: A061336 (x=2), A104246 (x=3), A283365 (x=4), A283370 (x=5).
x-simplex numbers: A000217 (x=2), A000292 (x=3), A000332 (x=4), A000389 (x=5), A000579 (x=6), A000580 (x=7), A000581 (x=8), A000582 (x=9).
Previous Showing 31-34 of 34 results.

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