A280386 Number of ways to write n as the sum of two squares and a term of A000009.

1, 2, 3, 3, 4, 5, 4, 4, 4, 6, 5, 6, 5, 6, 5, 5, 5, 6, 7, 7, 6, 7, 7, 5, 4, 7, 7, 9, 5, 7, 8, 7, 6, 5, 9, 6, 8, 8, 6, 10, 6, 9, 7, 8, 5, 7, 10, 7, 5, 6, 9, 9, 7, 10, 11, 10, 6, 9, 8, 5, 5, 8, 10, 10, 6, 8, 10, 10, 7, 8, 9, 10, 8, 8, 8, 9, 10, 7, 8, 11

Offset: 1

Author

Zhi-Wei Sun, Jan 01 2017

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 0.

Note that the main term of log A000009(n) is Pi*sqrt(n/3). So, A000009(n) eventually grows faster than any polynomial.

The conjecture was verified by the author for n up to 4*10^6. After learning this conjecture from the author, Prof. Qing-Hu Hou at Tianjin Univ. finished his verification of the above conjecture for n up to 10^9. - Zhi-Wei Sun, Jan 02 2017

Examples

			a(1) = 1 since 1 = 0^2 + 0^2 + 1 with 1 = A000009(1) = A000009(2).
a(2) = 2 since 2 = 0^2 + 1^2 + 1 = 0^2 + 0^2 + 2 with 1 = A000009(1) = A000009(2) and 2 = A000009(3) = A000009(4).
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28--Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Crossrefs

Cf. A000009, A000290, A001481.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
q[n_]:=q[n]=PartitionsQ[n];
ex={};Do[r=0;m=2;Label[bb];If[q[m]>n,Goto[cc]];Do[If[SQ[n-q[m]-x^2],r=r+1],{x,0,Sqrt[(n-q[m])/2]}];m=m+If[m<3,2,1];Goto[bb];Label[cc];ex=Append[ex,r];Continue,{n,1,80}]