0,2

In April 2009, Zhi-Wei Sun conjecturted that a(n)>0 for every n=0,1,2,3,.... Note that pentagonal numbers and hexagonal numbers are more sparse than squares and that there are infinitely many positive integers which cannnot be written as the sum of three squares.

On August 12, 2009, Zhi-Wei Sun made the following general conjecture on diagonal representations by polygonal numbers: For each integer m>2, any natural number n can be written in the form p_{m+1}(x_1)+...+p_{2m}(x_m) with x_1,...,x_m nonnegative integers, where p_k(x)=(k-2)x(x-1)/2+x (x=0,1,2,...) are k-gonal numbers. Sun has verified this with m=3 for n up to 10^6, and with m=4,5,6,7,8,9,10 for n up to 5*10^5. [From Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Aug 15 2009]

On August 21, 2009, Zhi-Wei Sun formulated the following strong version for his conjecture on diagonal representations by polygonal numbers: For any integer m>2, each natural number n can be expressed as p_{m+1}(x_1)+p_{m+2}(x_2)+p_{m+3}(x_3)+r with x_1,x_2,x_3 nonnegative integers and r an integer among 0,...,m-3. For m=3 and m=4,5,6,7,8,9,10, Sun has verified this conjecture for n up to 10^6 and 5*10^5 respectively. Sun also guessed that for each m=3,4,... all sufficiently large integers have the form p_{m+1}(x_1)+p_{m+2}(x_2)+p_{m+3}(x_3) with x_1,x_2,x_3 nonnegative integers. For example, it seems that 387904 is the largest integer not in the form p_{20}(x_1)+p_{21}(x_2)+p_{22}(x_3). [From Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Aug 21 2009]

On Sept. 4, 2009, Zhi-Wei Sun conjectured that the sequence contains every positive integer. For n=1,2,3,... let s(n) denote the least nonnegative integer m such that a(m)=n. Here is the list of s(1),...,s(30): 0, 9, 1, 6, 16, 36, 50, 37, 66, 82, 167, 121, 162, 236, 226, 276, 302, 446, 478, 532, 457, 586, 677, 521, 666, 852, 976, 877, 1006, 1046. [From Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Sep 04 2009]

M. B. Nathanson, A short proof of Cauchy's polygonal number theorem, Proc. Amer. Math. Soc. 99(1987), 22-24.

G. Pall, Large positive integers are sums of four or five values of a quadratic function, Amer. J. Math. 54(1932), 66-78.

Zhi-Wei Sun, On universal sums of polygonal numbers, preprint, arXiv:0905.0635. http://arxiv.org/abs/0905.0635

Zhi-Wei Sun, Table of n, a(n) for n = 0..50000

Zhi-Wei Sun, Various new conjectures involving polygonal numbers and primes (a message to Number Theory List), May 2009.

Zhi-Wei Sun, Mixed Sums of Primes and Other Terms (a webpage).

a(n)=|{<x,y,z>: x,y,z=0,1,2,... & x^2+(3y^2-y)/2+(2z^2-z)=n}|

For n=10 the a(10)=4 solutions are 4+0+6, 4+5+1, 9+0+1, 9+1+0.

SQ[x_]:=x>-1&&IntegerPart[Sqrt[x]]^2==x RN[n_]:=Sum[If[SQ[n-(3y^2-y)/2-(2z^2-z)], 1, 0], {y, 0, Sqrt[n]}, {z, 0, Sqrt[Max[0, n-(3y^2-y)/2]]}] Do[Print[n, " ", RN[n]], {n, 0, 50000}]

A000290, A000326, A000384

Sequence in context: A084101 A053386 A090569 this_sequence A109439 A133333 A133332

Adjacent sequences: A160321 A160322 A160323 this_sequence A160325 A160326 A160327

nonn

Zhi-Wei Sun (zwsun(AT)nju.edu.cn), May 08 2009