

A299537


Number of ways to write n^2 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that x or y is a power of 4 (including 4^0 = 1) and x + 3*y is also a power of 4.


31



1, 1, 1, 1, 1, 4, 1, 1, 4, 3, 1, 1, 2, 6, 1, 1, 2, 3, 1, 1, 8, 6, 2, 4, 3, 8, 3, 1, 6, 8, 4, 1, 6, 10, 3, 4, 2, 5, 6, 3, 4, 8, 1, 1, 7, 5, 1, 1, 5, 6, 4, 2, 4, 13, 5, 6, 7, 5, 5, 1, 3, 7, 2, 1, 3, 12, 6, 2, 11, 5, 5, 3, 7, 11, 2, 1, 6, 13, 5, 1
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OFFSET

1,6


COMMENTS

Conjecture (i): a(n) > 0 for all n > 0, and a(n) = 1 only for n = 4^k*m with k = 0,1,2,... and m = 1, 2, 3, 5, 7, 11, 15, 19, 43, 47, 135, 1103.
Conjecture (ii): For any integer n > 1, we can write n^2 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that 2*x or 2*y is a power of 4 and 2*(x+3*y) is also a power of 4.
Note that 81503^2 cannot be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and both x and x + 3*y in the set {4^k: k = 0,1,2,...}. However, 81503^2 = 16372^2 + 4^2 + 52372^2 + 60265^2 with 4 = 4^1 and 16372 + 3*4 = 4^7.
We have verified that the conjecture for n up to 10^7.
See also the related comments in A300219 and A300360, and a similar conjecture in A299794.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..2000
ZhiWei Sun, Refining Lagrange's foursquare theorem, J. Number Theory 175(2017), 167190.
ZhiWei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 20172018.


EXAMPLE

a(2) = 1 since 2^2 = 1^2 + 1^2 + 1^2 + 1^2 with 1 = 4^0 and 1 + 3*1 = 4^1.
a(5) = 1 since 5^2 = 4^2 + 0^2 + 0^2 + 3^2 with 4 = 4^1 and 4 + 3*0 = 4^1.
a(19) = 1 since 19^2 = 1^2 + 0^2 + 6^2 + 18^2 with 1 = 4^0 and 1 + 3*0 = 4^0.
a(43) = 1 since 43^2 = 4^2 + 20^2 + 8^2 + 37^2 with 4 = 4^1 and 4 + 3*20 = 4^3.
a(135) = 1 since 135^2 = 16^2 + 16^2 + 17^2 + 132^2 with 16 = 4^2 and 16 + 3*16 = 4^3.
a(1103) = 1 since 1103^2 = 4^2 + 4^2 + 716^2 + 839^2 with 4 = 4^1 and 4 + 3*4 = 4^2.


MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Pow[n_]:=Pow[n]=IntegerQ[Log[4, n]];
tab={}; Do[r=0; Do[If[(Pow[y]Pow[4^k3y])&&SQ[n^2y^2(4^k3y)^2z^2], r=r+1], {k, 0, Log[4, Sqrt[10]*n]}, {y, 0, Min[n, 4^k/3]}, {z, 0, Sqrt[Max[0, (n^2y^2(4^k3y)^2)/2]]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]


CROSSREFS

Cf. A000118, A000290, A000302, A271518, A279612, A281976, A299794, A299924, A300219, A300360, A300362.
Sequence in context: A070085 A131776 A010322 * A053239 A046569 A046596
Adjacent sequences: A299534 A299535 A299536 * A299538 A299539 A299540


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Mar 04 2018


STATUS

approved



