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.
%I A356037 #22 Oct 11 2022 14:10:28
%S A356037 1,3,5,8,10,13,15,15,19,24
%N A356037 Conjecturally, a(n) is the smallest number m such that every natural number is a sum of at most m n-simplex numbers.
%C A356037 n-simplex numbers are {binomial(k,n); k>=n}.
%C A356037 This problem is the simplex number analog of Waring's problem.
%C A356037 a(2) = 3 was proposed by Fermat and proved by Gauss, see A061336.
%C A356037 Pollock conjectures that a(3) = 5. Salzer and Levine prove this for numbers up to 452479659. See A104246 and A000797.
%C A356037 Kim gives a(4)=8, a(5)=10, a(6)=13 and a(7)=15 (not proved).
%H A356037 Hyun Kwang Kim, <a href="https://doi.org/10.1090/S0002-9939-02-06710-2">On regular polytope numbers</a>, Proc. Amer. Math. Soc. 131 (2003), p. 65-75.
%e A356037 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.
%e A356037 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.
%Y A356037 Cf. A002804, A079611.
%Y A356037 Minimal number of x-simplex numbers whose sum equals n: A061336 (x=2), A104246 (x=3), A283365 (x=4), A283370 (x=5).
%Y A356037 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).
%K A356037 nonn,hard,more
%O A356037 1,2
%A A356037 _Mohammed Yaseen_, Jul 24 2022