A165141 The least positive integer that can be written in exactly n ways as the sum of a square, a pentagonal number and a hexagonal number

3, 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, 1277, 1381, 1857, 1556, 1507, 1657, 1832, 1732, 2336, 2299, 2007, 2677, 2326, 2117, 2591, 2502, 2516, 2592, 3106, 3557

Offset

1,1

Comments

On Sep 04 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).

Links

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

F. Ge and Z. W. Sun, On some universal sums of generalized polygonal numbers, preprint, arXiv:0906.2450 [math.NT], 2009-2016.

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

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).

Zhi-Wei Sun, On universal sums of polygonal numbers, preprint, arXiv:0905.0635 [math.NT], 2009-2015.

Formula

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,...}.

Example

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.

Mathematica

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}]

Crossrefs

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

Sequence in context: A119796 A154572 A333346 * A019817 A243526 A329214

Adjacent sequences: A165138 A165139 A165140 * A165142 A165143 A165144

Keyword

nice,nonn

Author

Zhi-Wei Sun, Sep 05 2009

Status

approved