A302982 Number of ways to write n as x^2 + 5*y^2 + 2^z + 3*2^w with x,y,z,w nonnegative integers.

0, 0, 0, 1, 2, 1, 2, 4, 3, 3, 5, 4, 6, 7, 4, 7, 5, 4, 7, 8, 5, 5, 8, 5, 9, 7, 6, 13, 10, 7, 9, 10, 7, 12, 11, 8, 11, 7, 7, 11, 11, 6, 11, 13, 6, 10, 7, 7, 17, 13, 6, 13, 14, 9, 11, 18, 10, 13, 14, 11

Offset

1,5

Comments

Conjecture: a(n) > 0 for all n > 3.

Clearly, a(4*n) > 0 if a(n) > 0. We have verified a(n) > 0 for all n = 4..2*10^8.

See also A302983 and A302984 for similar conjectures.

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.

Example

a(4) = 1 with 4 = 0^2 + 5*0^2 + 2^0 + 3*2^0.
a(5) = 2 with 5 =  1^2 + 5*0^2 + 2^0 + 3*2^0 = 0^2 + 5*0^2 + 2^1 + 3*2^0.
a(6) = 1 with 6 = 1^2 + 3*0^2 + 2^1 + 3*2^0.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[n-3*2^k-2^j-5x^2], r=r+1], {k, 0, Log[2, n/3]}, {j, 0, If[n==3*2^k, -1, Log[2, n-3*2^k]]}, {x, 0, Sqrt[(n-3*2^k-2^j)/5]}]; tab=Append[tab, r], {n, 1, 60}]; Print[tab]

Crossrefs

Cf. A000079, A000290, A020669, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A301534, A302920, A302981, A302983, A302984.

Sequence in context: A209128 A209131 A165053 * A238577 A131380 A100461

Adjacent sequences: A302979 A302980 A302981 * A302983 A302984 A302985

Keyword

nonn

Author

Zhi-Wei Sun, Apr 16 2018

Status

approved