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 A275678 Number of ordered ways to write n as 4^k*(1+4*x^2+y^2) + z^2, where k,x,y,z are nonnegative integers with x <= y. 5
 1, 2, 1, 1, 3, 3, 1, 2, 3, 4, 2, 1, 2, 3, 2, 1, 4, 4, 1, 3, 5, 3, 1, 3, 5, 5, 3, 1, 2, 7, 2, 2, 5, 3, 3, 3, 6, 2, 2, 4, 6, 7, 1, 2, 4, 7, 1, 1, 3, 5, 5, 2, 5, 5, 4, 3, 8, 4, 2, 2, 1, 7, 3, 1, 6, 8, 2, 4, 8, 6, 2, 4, 6, 3, 4, 1, 3, 6, 2, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture: (i) a(n) > 0 for all n > 0. (ii) Any positive integer can be written as 4^k*(1+4*x^2+y^2) + z^2, where k,x,y,z are nonnegative integers with x <= z. This is stronger than Lagrange's four-square theorem. We have shown that each n = 1,2,3,... can be written as 4^k*(1+4*x^2+y^2) + z^2 with k,x,y,z nonnegative integers. See also A275656, A275675 and A275676 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, arXiv:1604.06723 [math.GM], 2016. EXAMPLE a(12) = 1 since 12 = 4*(1+4*0^2+1^2) + 2^2 with 0 < 1. a(19) = 1 since 19 = 4^0*(1+4*0^2+3^2) + 3^2 with 0 < 3. a(61) = 1 since 61 = 4*(1+4*1^2+2^2) + 5^2 with 1 < 2. a(125) = 1 since 125 = 4*(1+4*0^2+0^2) + 11^2 with 0 = 0. a(359) = 1 since 359 = 4^0*(1+4*7^2+9^2) + 9^2 with 7 < 9. a(196253) = 1 since 196253 = 4*(1+4*0^2+0^2) + 443^2 with 0 = 0. MATHEMATICA SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]] Do[r=0; Do[If[SQ[n-4^k*(1+4x^2+y^2)], r=r+1], {k, 0, Log[4, n]}, {x, 0, Sqrt[(n/4^k-1)/5]}, {y, x, Sqrt[n/4^k-1-4x^2]}]; Print[n, " ", r]; Continue, {n, 1, 80}] CROSSREFS Cf. A000118, A000290, A271518, A275648, A275656, A275675, A275676. Sequence in context: A056670 A170925 A030189 * A273108 A306405 A114162 Adjacent sequences:  A275675 A275676 A275677 * A275679 A275680 A275681 KEYWORD nonn AUTHOR Zhi-Wei Sun, Aug 05 2016 STATUS approved

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Last modified August 2 05:02 EDT 2021. Contains 346409 sequences. (Running on oeis4.)