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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 16 2021
STATUS
approved