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

Author

Zhi-Wei Sun, Apr 16 2021

Keywords

• nonn

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

Examples

			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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000079, A000578, A001014, A001318, A343397, A343411.

Programs

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]