A302984 Number of ways to write n as x^2 + 2*y^2 + 2^z + 5*2^w with x,y,z,w nonnegative integers.

0, 0, 0, 0, 0, 1, 2, 2, 3, 3, 3, 4, 5, 5, 8, 5, 5, 7, 4, 6, 7, 9, 9, 10, 10, 7, 9, 8, 10, 15, 10, 9, 10, 8, 6, 10, 10, 11, 14, 14, 8, 12, 13, 13, 20, 15, 12, 16, 10, 15, 12, 10, 15, 17, 16, 12, 16, 14, 14, 21

Offset

1,7

Comments

Conjecture: a(n) > 0 for all n > 5.

Clearly, a(2*n) > 0 if a(n) > 0. We have verified a(n) > 0 for all n = 6...10^9.

See also A302982 and A302983 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(6) = 1 with 6 = 0^2 + 2*0^2 + 2^0 + 5*2^0.
a(7) = 2 with 7 = 1^2 + 2*0^2 + 2^0 + 5*2^0 = 0^2 + 2*0^2 + 2^1 + 5*2^0.
a(8) = 2 with 8 = 0^2 + 2*1^2 + 2^0 + 5*2^0 = 1^2 + 2*0^2 + 2^1 + 5*2^0.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[MemberQ[{5, 7}, Mod[Part[Part[f[n], i], 1], 8]]&&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-5*2^k-2^j], Do[If[SQ[n-5*2^k-2^j-2x^2], r=r+1], {x, 0, Sqrt[(n-5*2^k-2^j)/2]}]], {k, 0, Log[2, n/5]}, {j, 0, Log[2, Max[1, n-5*2^k]]}]; tab=Append[tab, r], {n, 1, 60}]; Print[tab]

Crossrefs

Cf. A000079, A000290, A002479, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302985.

Sequence in context: A369611 A029112 A348336 * A029094 A262950 A227398

Adjacent sequences: A302981 A302982 A302983 * A302985 A302986 A302987

Keyword

nonn

Author

Zhi-Wei Sun, Apr 16 2018

Status

approved