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 A301452 Number of ways to write n^2 as m*4^k + x^2 + 2*y^2 with m in the set {2, 3} and k,x,y nonnegative integers. 11
 0, 2, 2, 2, 2, 5, 3, 2, 4, 4, 4, 5, 5, 5, 6, 2, 4, 6, 5, 4, 9, 5, 4, 5, 5, 7, 10, 5, 6, 7, 8, 2, 6, 6, 7, 6, 9, 7, 10, 4, 6, 12, 3, 5, 10, 5, 6, 5, 5, 8, 9, 7, 7, 12, 5, 5, 13, 9, 6, 7, 8, 10, 13, 2, 6, 8, 10, 6, 15, 9, 9, 6, 10, 9, 12, 7, 8, 13, 6, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture: a(n) > 0 for all n > 1. We call this the 2-3 conjecture. It is simialr to the author's 2-5 conjecture which states that A300510(n) > 0 for all n > 1. We have verified that a(n) > 0 for all n = 2..5*10^7. It is known that the number of ways to write a positive integer n as x^2 + 2*y^2 with x and y integers is twice the difference |{d > 0: d|n and d == 1,3 (mod 8)| - |{d>0: d|n and d == 5,7 (mod 8)}|. 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) = 2 since 2^2 = 2*4^0 + 0^2 + 2*1^2 and 2^2 = 3*4^0 + 1^2 + 2*0^2. a(3) = 2 since 3^2 = 2*4^1 + 1^2 + 2*0^2 and 3^2 = 3*4^0 + 2^2 + 2*1^2. a(5) = 2 since 5^2 = 2*4^1 + 3^2 + 2*2^2 and 5^2 = 3*4^0 + 2^2 + 2*3^2. MATHEMATICA f[n_]:=f[n]=n/2^(IntegerExponent[n, 2]); OD[n_]:=OD[n]=Divisors[f[n]]; QQ[n_]:=QQ[n]=(n==0)||(n>0&&Sum[JacobiSymbol[-2, Part[OD[n], i]], {i, 1, Length[OD[n]]}]!=0); SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; tab={}; Do[r=0; Do[If[QQ[n^2-m*4^k], Do[If[SQ[n^2-m*4^k-2x^2], r=r+1], {x, 0, Sqrt[(n^2-m*4^k)/2]}]], {m, 2, 3}, {k, 0, Log[4, n^2/m]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab] CROSSREFS Cf. A000290, A000302, A002479, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391. Sequence in context: A080647 A324516 A181058 * A322788 A177333 A118486 Adjacent sequences:  A301449 A301450 A301451 * A301453 A301454 A301455 KEYWORD nonn AUTHOR Zhi-Wei Sun, Mar 21 2018 STATUS approved

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Last modified December 14 19:27 EST 2019. Contains 329987 sequences. (Running on oeis4.)