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

1,2

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

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.

Example

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).

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}]

Crossrefs

Cf. A000009, A000290, A001481.

Sequence in context: A325784 A244929 A302920 * A204979 A243351 A376635

Adjacent sequences: A280383 A280384 A280385 * A280387 A280388 A280389

Keyword

nonn

Author

Zhi-Wei Sun, Jan 01 2017

Status

approved