A347827 Number of ways to write n as w^4 + x^4 + (y^2 + 23*z^2)/16, where w is zero or a power of two (including 2^0 = 1), and x,y,z are nonnegative integers.

1, 3, 4, 4, 4, 3, 2, 2, 2, 4, 5, 2, 2, 5, 4, 1, 4, 8, 8, 8, 6, 2, 2, 3, 6, 12, 9, 5, 9, 9, 4, 2, 8, 8, 6, 5, 4, 6, 4, 4, 11, 11, 6, 7, 6, 3, 3, 5, 11, 8, 7, 3, 9, 10, 5, 11, 9, 3, 4, 5, 3, 2, 3, 7, 10, 10, 6, 2, 7, 5, 8, 10, 5, 9, 7, 6, 4, 1, 6, 9, 9, 9, 10, 7, 5, 4, 6, 5, 13, 11, 6, 5, 3, 6, 16, 11, 6, 15, 15, 7, 5

Offset

0,2

Comments

23-Conjecture: a(n) > 0 for all n = 0,1,2,....

This is stronger than the conjecture in A347824, and it has been verified for n up to 3*10^6. See also a similar conjecture in A347562.

It seems that a(n) = 1 only for n = 0, 15, 77, 231, 291, 437, 471, 1161, 1402, 4692, 7107, 9727.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Example

a(231) = 1 with 231 = 0^4 + 3^4 + (10^2 + 23*10^2)/16.
a(437) = 1 with 437 = 3^4 + 4^4 + (40^2 + 23*0^2)/16.
a(1402) = 1 with 1402 = 2^4 + 5^4 + (3^2 + 23*23^2)/16.
a(9727) = 1 with 9727 = 0^4 + 6^4 + (367^2 + 23*3^2)/16.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; tab={}; Do[r=0; Do[If[w==0||IntegerQ[Log[2, w]], Do[If[SQ[16(n-w^4-x^4)-23z^2], r=r+1], {x, 0, (n-w^4)^(1/4)}, {z, 0, Sqrt[16(n-w^4-x^4)/23]}]], {w, 0, n^(1/4)}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]

Crossrefs

Cf. A000079, A000290, A000583, A347562, A347824.

Sequence in context: A199185 A279781 A262827 * A143490 A007485 A280356

Adjacent sequences: A347824 A347825 A347826 * A347828 A347829 A347830

Keyword

nonn

Author

Zhi-Wei Sun, Jan 23 2022

Status

approved