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Conjecturally, a(n) is the smallest number m such that every natural number is a sum of at most m n-simplex numbers.
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%I #22 Oct 11 2022 14:10:28

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

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

%C n-simplex numbers are {binomial(k,n); k>=n}.

%C This problem is the simplex number analog of Waring's problem.

%C a(2) = 3 was proposed by Fermat and proved by Gauss, see A061336.

%C Pollock conjectures that a(3) = 5. Salzer and Levine prove this for numbers up to 452479659. See A104246 and A000797.

%C Kim gives a(4)=8, a(5)=10, a(6)=13 and a(7)=15 (not proved).

%H 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 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 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 Cf. A002804, A079611.

%Y Minimal number of x-simplex numbers whose sum equals n: A061336 (x=2), A104246 (x=3), A283365 (x=4), A283370 (x=5).

%Y 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 nonn,hard,more

%O 1,2

%A _Mohammed Yaseen_, Jul 24 2022