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 A303338 Number of ways to write n as x^2 + 2*y^2 + 3*2^z + 4^w with x,y,z,w nonnegative integers. 32
 0, 0, 0, 1, 1, 1, 3, 3, 2, 4, 3, 2, 6, 2, 4, 8, 2, 4, 7, 3, 4, 8, 5, 5, 10, 6, 4, 10, 8, 5, 12, 7, 3, 12, 4, 5, 12, 5, 5, 14, 7, 4, 12, 7, 6, 12, 6, 6, 10, 7, 7, 12, 7, 6, 14, 6, 8, 16, 4, 8, 18, 5, 6, 16, 5, 9, 13, 7, 7, 14 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS Conjecture: a(n) > 0 for all n > 3. This is stronger than the author's previous conjecture in A302983. I call it the 1-2-3-4 conjecture. It has been verified that a(n) > 0 for all n = 4..10^9. 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, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120. Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018. EXAMPLE a(4) = 1 with 4 = 0^2 + 2*0^2 + 3*2^0 + 4^0. a(5) = 1 with 5 = 1^2 + 2*0^2 + 3*2^0 + 4^0. a(6) = 1 with 6 = 0^2 + 2*1^2 + 3*2^0 + 4^0. a(9) = 2 with 9 = 0^2 + 2*1^2 + 3*2^0 + 4^1 = 0^2 + 2*1^2 + 3*2^1 + 4^0. MATHEMATICA f[n_]:=f[n]=FactorInteger[n]; g[n_]:=g[n]=Sum[Boole[MemberQ[{5, 7}, Mod[Part[Part[f[n], i], 1], 8]]&&Mod[Part[Part[f[n], i], 2], 2]==1], {i, 1, Length[f[n]]}]==0; QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]); SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; tab={}; Do[r=0; Do[If[QQ[n-3*2^k-4^j], Do[If[SQ[n-3*2^k-4^j-2x^2], r=r+1], {x, 0, Sqrt[(n-3*2^k-4^j)/2]}]], {k, 0, Log[2, n/3]}, {j, 0, If[3*2^k==n, -1, Log[4, n-3*2^k]]}]; tab=Append[tab, r], {n, 1, 70}]; Print[tab] CROSSREFS Cf. A000079, A000290, A002479, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303363. Sequence in context: A247509 A106686 A106702 * A271510 A282545 A306471 Adjacent sequences:  A303335 A303336 A303337 * A303339 A303340 A303341 KEYWORD nonn AUTHOR Zhi-Wei Sun, Apr 22 2018 STATUS approved

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Last modified March 18 22:11 EDT 2019. Contains 321305 sequences. (Running on oeis4.)