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A300139 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 4*x - 3*y a square, where x,y,z,w are nonnegative integers with z <= w such that 10*x or y is a square. 1
1, 2, 3, 2, 2, 2, 2, 1, 1, 2, 4, 3, 2, 1, 2, 2, 2, 3, 5, 3, 4, 2, 1, 1, 1, 4, 6, 5, 2, 3, 3, 1, 3, 4, 5, 4, 5, 3, 3, 2, 2, 6, 6, 2, 1, 4, 2, 2, 2, 2, 9, 6, 6, 3, 4, 3, 1, 4, 3, 4, 4, 4, 3, 3, 2, 6, 9, 4, 5, 4, 4, 1, 2, 4, 7, 9, 2, 3, 3, 1, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Conjecture 1: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 16^k*m with k = 0,1,2,... and m = 0, 7, 8, 13, 22, 23, 24, 31, 44, 56, 71, 79, 88, 109, 120, 152, 184, 472, 1912, 6008, 9080.
Conjecture 2: Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with 3*x - y twice a square, where x,y,z,w are nonnegative integers such that 5*x or y is a square.
By the author's 2017 JNT paper, any nonnegative integer can be written as the sum of a fourth power and three squares.
See also A281976, A300666, A300667, A300708 and A300712 for similar conjectures.
a(n) > 0 for all n = 0..10^8. Also, Conjecture 2 holds for all n = 0..10^8. - Zhi-Wei Sun, Oct 05 2020
LINKS
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(22) = 1 since 22 = 1^2 + 1^2 + 2^2 + 4^2 with 1 = 1^2 and 4*1 - 3*1 = 1^2.
a(23) = 1 since 23 = 3^2 + 1^2 + 2^2 + 3^2 with 1 = 1^2 and 4*3 - 3*1 = 3^2.
a(109) = 1 since 109 = 0^2 + 0^2 + 3^2 + 10^2 with 0 = 0^2 and 4*0 - 3*0 = 0^2.
a(184) = 1 since 184 = 10^2 + 8^2 + 2^2 + 4^2 with 10*10 = 10^2 and 4*10 - 3*8 = 4^2.
a(6008) = 1 since 6008 = 12^2 + 16^2 + 42^2 + 62^2 with 16 = 4^2 and 4*12 - 3*16 = 0^2.
a(9080) = 1 since 9080 = 10^2 + 12^2 + 0^2 + 94^2 with 10*10 = 10^2 and 4*10 - 3*12 = 2^2.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[Mod[m^2+3y, 4]==0&&(SQ[10(m^2+3y)/4]||SQ[y]), Do[If[SQ[n-((m^2+3y)/4)^2-y^2-z^2], r=r+1], {z, 0, Sqrt[Max[0, (n-((m^2+3y)/4)^2-y^2)/2]]}]], {m, 0, 2n^(1/4)}, {y, 0, 4/5*Sqrt[n-m^4/16]}]; tab=Append[tab, r], {n, 0, 80}]; Print[tab]
CROSSREFS
Sequence in context: A128830 A090387 A030329 * A300666 A120881 A297930
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 12 2018
STATUS
approved

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Last modified September 4 21:56 EDT 2024. Contains 375685 sequences. (Running on oeis4.)