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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.
3
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, 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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 23 2022
STATUS
approved