1,1

On Sept. 4, 2009 Zhi-Wei Sun conjectured that the sequence A160324 contains every positive integer, i.e., for each positive integer n there exists a positive integer s which can be written in exactly n ways as the sum of a square, a pentagonal number and a hexagonal number. Based on this conjecture we create the current sequence. It seems that 0.9<a(n)/n^2<1.6 for n>33. Zhi-Wei Sun conjectured that a(n)/n^2 has a limit c with 1.1<c<1.2. On Sun's request, his friend Qing-Hu Hou produced a list of a(n) for n=1,...,913 (see the b-file).

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

Z. W. Sun, On universal sums of polygonal numbers, preprint, arXiv:0905.0635. http://arxiv.org/abs/0905.0635

F. Ge and Z. W. Sun, On some universal sums of generalized polygonal numbers, preprint, arXiv:0906.2450. http://arxiv.org/abs/0906.2450

Zhi-Wei Sun, Table of n, a(n) for n = 1..913

Zhi-Wei Sun, A challenging conjecture on sums of polygonal number (a message to Number Theory List), 2009.

Zhi-Wei Sun, Polygonal numbers, primes and ternary quadratic forms (a talk given at a number theory conference), 2009.

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

a(n)=min{m>0: m=x^2+(3y^2-y)/2+(2z^2-z) has exactly n solutions with x,y,z=0,1,2,...}

For n=5 the a(5)=16 solutions are 0^2+1+15=1^2+0+15=2^2+12+0=3^2+1+6=4^2+0+0=16.

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[Do[If[RN[m]==n, Print[n, " ", m]; Goto[aa]], {m, 1, 1000000}]; Label[aa]; Continue, {n, 1, 100}]

Cf. A160324, A000290, A000326, A000384, A160325, A160326

Sequence in context: A113847 A119796 A154572 this_sequence A019817 A080322 A126179

Adjacent sequences: A165138 A165139 A165140 this_sequence A165142 A165143 A165144

nice,nonn

Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Sep 05 2009