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 A275675 Number of ordered ways to write n as 4^k*(1+x^2+y^2)+5*z^2, where k,x,y,z are nonnegative integers with x <= y. 7
 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 3, 2, 2, 3, 2, 4, 1, 1, 2, 2, 3, 3, 1, 1, 2, 2, 3, 3, 1, 3, 3, 2, 1, 2, 1, 5, 3, 2, 2, 3, 4, 1, 4, 2, 3, 5, 2, 2, 3, 1, 3, 3, 1, 4, 2, 4, 1, 2, 3, 2, 6, 2, 3, 3, 2, 2, 2, 2, 2, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Conjecture: a(n) > 0 for all n > 0. This is stronger than the known fact that any natural number can be written as w^2 + x^2 + y^2 + 5*z^2 with w,x,y,z integers. See also A275656, A275676 and A275678 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(43) = 1 since 43 = 4^0*(1+1^2+6^2) + 5*1^2. a(45) = 1 since 45 = 4*(1+0^2+3^2) + 5*1^2. a(237) = 1 since 237 = 4^3*(1+1^2+1^2) + 5*3^2. a(561) = 1 since 561 = 4*(1+8^2+8^2) + 5*3^2. a(9777) = 1 since 9777 = 4*(1+11^2+31^2) + 5*33^2. a(39108) = 1 since 39108 = 4^2*(1+11^2+31^2) + 5*66^2. MATHEMATICA SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]] Do[r=0; Do[If[SQ[(n-4^k*(1+x^2+y^2))/5], r=r+1], {k, 0, Log[4, n]}, {x, 0, Sqrt[(n/4^k-1)/2]}, {y, x, Sqrt[n/4^k-1-x^2]}]; Print[n, " ", r]; Continue, {n, 1, 80}] CROSSREFS Cf. A000118, A000290, A271518, A275648, A275656, A275676, A275678. Sequence in context: A120256 A300647 A114811 * A219025 A270746 A043531 Adjacent sequences:  A275672 A275673 A275674 * A275676 A275677 A275678 KEYWORD nonn AUTHOR Zhi-Wei Sun, Aug 04 2016 STATUS approved

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Last modified January 23 07:07 EST 2020. Contains 331168 sequences. (Running on oeis4.)