

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
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OFFSET

1,3


COMMENTS

Conjecture: a(n) > 0 for all n > 1.
We have verified a(n) > 0 for n = 2..10^7.


LINKS



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[nx^6y^32^k], r=r+1], {x, 0, (n1)^(1/6)}, {y, 0, (nx^61)^(1/3)}, {k, 1, Log[2, nx^6y^3]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



