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 A343460 Number of ways to write n as x^6 + y^3 + z*(3*z+1)/2 + 2^k, where x and y are nonnegative integers, z is an integer and k is a positive integer. 4
 0, 1, 3, 5, 6, 5, 4, 4, 6, 9, 8, 6, 5, 5, 6, 7, 11, 11, 7, 5, 5, 5, 5, 8, 8, 5, 4, 5, 7, 7, 10, 11, 7, 8, 8, 8, 8, 9, 10, 8, 6, 7, 10, 10, 10, 7, 6, 7, 4, 5, 7, 6, 5, 4, 7, 8, 6, 5, 7, 8, 7, 6, 3, 5, 8, 12, 15, 13, 12, 10, 9, 11, 17, 18, 13, 9, 6, 9, 11, 16 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Conjecture: a(n) > 0 for all n > 1. We have verified a(n) > 0 for n = 2..10^7. Conjecture verified up to 10^10. - Giovanni Resta, Apr 18 2021 LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 EXAMPLE a(2) = 1 with 2 = 0^6 + 0^3 + 0*(3*0+1)/2 + 2^1. a(175) = 2 with 175 = 1^6 + 3^3 + (-10)*(3*(-10)+1)/2 + 2^1 = 2^6 + 4^3 + 3*(3*3+1)/2 + 2^5. a(14553) = 1 with 14553 = 2^6 + 17^3 + (-80)*(3*(-80)+1)/2 + 2^4. MATHEMATICA PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]]; tab={}; Do[r=0; Do[If[PenQ[n-x^6-y^3-2^k], r=r+1], {x, 0, (n-1)^(1/6)}, {y, 0, (n-x^6-1)^(1/3)}, {k, 1, Log[2, n-x^6-y^3]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab] CROSSREFS Cf. A000079, A000578, A001014, A001318, A343397, A343411. Sequence in context: A110279 A161435 A354213 * A224831 A281591 A267884 Adjacent sequences: A343457 A343458 A343459 * A343461 A343462 A343463 KEYWORD nonn AUTHOR Zhi-Wei Sun, Apr 16 2021 STATUS approved

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Last modified August 10 18:57 EDT 2024. Contains 375058 sequences. (Running on oeis4.)