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 A303656 Number of ways to write n as a^2 + b^2 + 3^c + 5^d, where a,b,c,d are nonnegative integers with a <= b. 22
 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, 5, 6, 4, 6, 5, 5, 6, 6, 5, 8, 4, 6, 6, 5, 4, 7, 5, 7, 5, 6, 4, 5, 3, 4, 7, 6, 7, 8, 5, 4, 7, 5, 5, 9, 3, 6, 5, 6, 4, 6, 5, 7, 7, 4, 5, 5, 5, 4, 6, 5, 6, 10, 5, 4, 5, 7, 4, 9, 2, 9, 8, 5, 6, 6 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Conjecture: a(n) > 0 for all n > 1. In other words, any integer n > 1 can be written as the sum of two squares, a power of 3 and a power of 5. It has been verified that a(n) > 0 for all n = 2..2*10^10. It seems that any integer n > 1 also can be written as the sum of two squares, a power of 2 and a power of 3. The author would like to offer 3500 US dollars as the prize for the first proof of his conjecture that a(n) > 0 for all n > 1. - Zhi-Wei Sun, Jun 05 2018 LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..100000 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(2) = 1 with 2 = 0^2 + 0^2 + 3^0 + 5^0. a(5) = 1 with 5 = 0^2 + 1^2 + 3^1 + 5^0. a(25) = 1 with 25 = 1^2 + 4^2 + 3^1 + 5^1. 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], Do[If[SQ[n-3^k-5^m-x^2], r=r+1], {x, 0, Sqrt[(n-3^k-5^m)/2]}]], {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, A303429, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303702, A303821. Sequence in context: A007828 A070804 A303429 * A253630 A104481 A078709 Adjacent sequences:  A303653 A303654 A303655 * A303657 A303658 A303659 KEYWORD nonn AUTHOR Zhi-Wei Sun, Apr 27 2018 STATUS approved

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Last modified October 19 13:01 EDT 2019. Contains 328222 sequences. (Running on oeis4.)