A303429 Number of ordered pairs (k, m) of nonnegative integers such that n - 3^k - 5^m can be written as the sum of two squares.

0, 1, 1, 2, 1, 3, 2, 3, 2, 4, 3, 4, 2, 4, 4, 3, 2, 4, 4, 3, 2, 4, 3, 4, 1, 4, 4, 6, 3, 6, 4, 5, 5, 6, 4, 8, 4, 6, 5, 5, 4, 7, 5, 7, 5, 6, 4, 5, 3, 4, 6, 5, 5, 7, 5, 3, 6, 4, 4, 8, 3, 6, 5, 5, 4, 6, 4, 7, 6, 4, 4, 5, 4, 4, 5, 4, 5, 8, 4, 4, 5, 6, 4, 8, 2, 9, 7, 5, 5, 6

Offset

1,4

Comments

Conjecture: a(n) > 0 for all n > 1.

This is equivalent to the author's conjecture in A303656.

It has been verified that a(n) > 0 for all n = 2..10^9.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.

Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Maple

a(5) = 1 with 5 - 3^1 - 5^0 = 0^2 + 1^2.
a(25) = 1 with 25 - 3^1 - 5^1 = 1^2 + 4^2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n], i], 1], 4]==3&&Mod[Part[Part[f[n], i], 2], 2]==1], {i, 1, Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={}; Do[r=0; Do[If[QQ[n-3^k-5^m], r=r+1], {k, 0, Log[3, n]}, {m, 0, If[n==3^k, -1, Log[5, n-3^k]]}]; tab=Append[tab, r], {n, 1, 90}]; Print[tab]

Crossrefs

Cf. A000244, A000290, A000351, A001481, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656.

Sequence in context: A262746 A007828 A070804 * A303656 A253630 A362938

Adjacent sequences: A303426 A303427 A303428 * A303430 A303431 A303432

Keyword

nonn

Author

Zhi-Wei Sun, Apr 28 2018

Status

approved