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 A303429 Number of ordered pairs (k, m) of nonnegative integers such that n - 3^k - 5^m can be written as the sum of two squares. 2
 0, 1, 1, 2, 1, 3, 2, 3, 2, 4, 3, 4, 2, 4, 4, 3, 2, 4, 4, 3, 2, 4, 3, 4, 1, 4, 4, 6, 3, 6, 4, 5, 5, 6, 4, 8, 4, 6, 5, 5, 4, 7, 5, 7, 5, 6, 4, 5, 3, 4, 6, 5, 5, 7, 5, 3, 6, 4, 4, 8, 3, 6, 5, 5, 4, 6, 4, 7, 6, 4, 4, 5, 4, 4, 5, 4, 5, 8, 4, 4, 5, 6, 4, 8, 2, 9, 7, 5, 5, 6 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Conjecture: a(n) > 0 for all n > 1. This is equivalent to the author's conjecture in A303656. It has been verified that a(n) > 0 for all n = 2..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. MAPLE a(5) = 1 with 5 - 3^1 - 5^0 = 0^2 + 1^2. a(25) = 1 with 25 - 3^1 - 5^1 = 1^2 + 4^2. MATHEMATICA SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; f[n_]:=f[n]=FactorInteger[n]; g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n], i], 1], 4]==3&&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]); tab={}; Do[r=0; Do[If[QQ[n-3^k-5^m], r=r+1], {k, 0, Log[3, n]}, {m, 0, If[n==3^k, -1, Log[5, n-3^k]]}]; tab=Append[tab, r], {n, 1, 90}]; Print[tab] CROSSREFS Cf. A000244, A000290, A000351, A001481, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656. Sequence in context: A262746 A007828 A070804 * A303656 A253630 A104481 Adjacent sequences:  A303426 A303427 A303428 * A303430 A303431 A303432 KEYWORD nonn AUTHOR Zhi-Wei Sun, Apr 28 2018 STATUS approved

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Last modified June 1 12:52 EDT 2020. Contains 334762 sequences. (Running on oeis4.)