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A299794 Number of ways to write n^2 as x^2 + y^2 + z^2 + w^2 with x >= y >= 0 <= z <= w such that x or 2*y is a power of 4 (including 4^0 = 1) and x + 15*y is also a power of 4. 30
1, 1, 1, 1, 1, 3, 1, 1, 4, 2, 1, 1, 2, 3, 1, 1, 2, 2, 1, 1, 6, 3, 1, 3, 3, 2, 2, 1, 3, 4, 2, 1, 5, 4, 4, 5, 1, 2, 3, 2, 5, 5, 2, 2, 8, 2, 2, 1, 5, 2, 4, 3, 4, 4, 4, 3, 6, 3, 2, 3, 3, 4, 3, 1, 3, 6, 4, 3, 11, 2, 2, 2, 4, 5, 1, 2, 3, 5, 3, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Conjecture 1: a(n) > 0 for all n > 0. Also, for any integer n > 1 we can write n^2 as x^2 + y^2 + z^2 + w^2 with x >= y >= 0 <= z <= w such that 2*x or y is a power of 4 and also x + 15*y = 2^(2k+1) for some k = 0,1,2,....

Conjecture 2: Let d be 2 or 8, and let r be 0 or 1. Then any positive square n^2 can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x or y is a power of 2 and x + d*y = 2^(2k+r) for some k = 0,1,2,....

We have verified Conjecture 1 for n up to 10^7.

See also A299537, A300219 and A300396 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, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

EXAMPLE

a(2) = 1 since 2^2 = 1^2 + 1^2 + 1^2 + 1^2 with 1 = 4^0 and 1 + 15*1 = 4^2.

a(5) = 1 since 5^2 = 4^2 + 0^2 + 0^2 + 3^2 with 4 = 4^1 and 4 + 15*0 = 4^1.

a(19) = 1 since 19^2 = 1^2 + 0^2 + 6^2 + 18^2 with 1 = 4^0 and 1 + 15*0 = 4^0.

a(159) = 1 since 159^2 = 34^2 + 2^2 + 75^2 + 136^2 with 2*2 = 4^1 and 34 + 15*2 = 4^3.

a(1998) = 1 since 1998^2 = 256^2 + 256^2 + 286^2 + 1944^2 with 256 = 4^4 and 256 + 15*256 = 4^6.

a(3742) = 1 since 3742^2 = 2176^2 + 128^2 + 98^2 + 3040^2 with 2*128 = 4^4 and 2176 + 15*128 = 4^6.

MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

Pow[n_]:=Pow[n]=IntegerQ[Log[4, n]];

tab={}; Do[r=0; Do[If[Pow[2y]||Pow[4^k-15y], Do[If[SQ[n^2-y^2-(4^k-15y)^2-z^2], r=r+1], {z, 0, Sqrt[Max[0, (n^2-y^2-(4^k-15y)^2)/2]]}]],

{k, 0, Log[4, Sqrt[226]*n]}, {y, 0, Min[n, 4^(k-2)]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]

CROSSREFS

Cf. A000118, A000290, A000302, A271518, A281976, A299537, A299924, A300219, A300356, A300360, A300362, A300396.

Sequence in context: A191780 A098712 A264490 * A023579 A023577 A188139

Adjacent sequences:  A299791 A299792 A299793 * A299795 A299796 A299797

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Mar 04 2018

STATUS

approved

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Last modified March 21 04:59 EDT 2019. Contains 321364 sequences. (Running on oeis4.)