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.

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

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