A347562 Number of ways to write n as 16^w + x^2 + (y^2 + 23*z^4)/324, where w,x,y,z are nonnegative integers.

1, 2, 2, 1, 2, 2, 2, 1, 2, 3, 3, 1, 2, 4, 1, 2, 5, 6, 4, 3, 4, 3, 4, 2, 4, 7, 4, 5, 5, 4, 2, 4, 6, 5, 4, 3, 6, 8, 2, 1, 8, 7, 6, 2, 4, 6, 2, 3, 5, 7, 6, 7, 10, 4, 1, 6, 4, 7, 4, 2, 5, 6, 6, 4, 7, 9, 5, 7, 5, 1, 3, 2, 8, 6, 4, 1, 6, 7, 3, 4, 6, 6, 7, 5, 1, 7, 2, 7, 3, 6, 5, 1, 3, 5, 5, 3, 7, 11, 4, 2

Offset

1,2

Comments

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

This has been verified for n up to 10^6.

It seems that a(n) = 1 only for n = 1, 4, 8, 12, 15, 40, 55, 70, 76, 85, 92, 104, 135, 156, 177, 192, 204, 207, 231, 279, 300, 447, 567, 1427, 1887, 4371.

See also A347827 for a similar conjecture.

Links

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

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

Example

a(15) = 1 with 15 = 16^0 + 1^2 + (62^2 + 23*2^4)/324.
a(156) = 1 with 156 = 16^1 + 6^2 + (139^2 + 23*5^4)/324.
a(300) = 1 with 300 = 16^2 + 6^2 + (27^2 + 23*3^4)/324.
a(1427) = 1 with 1427 = 16^1 + 35^2 + (71^2 + 23*7^4)/324.
a(1887) = 1 with 1887 = 16^1 + 15^2 + (729^2 + 23*3^4)/324.
a(4371) = 1 with 4371 = 16^1 + 63^2 + (351^2 + 23*3^4)/324.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[324(n-16^w-x^2)-23y^4], r=r+1], {w, 0, Log[16, n]}, {x, 0, Sqrt[n-16^w]},
{y, 0, (324(n-16^w-x^2)/23)^(1/4)}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000290, A000583, A001025, A347824, A347827, A350860.

Sequence in context: A367816 A116543 A256911 * A359790 A107260 A338236

Adjacent sequences: A347559 A347560 A347561 * A347563 A347564 A347565

Keyword

nonn

Author

Zhi-Wei Sun, Jan 23 2022

Status

approved