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

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

Offset: 1

Author

Mohammed Yaseen, Jul 24 2022

Keywords

• nonn
• hard
• more

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

• Hyun Kwang Kim, On regular polytope numbers, Proc. Amer. Math. Soc. 131 (2003), p. 65-75.

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).