A308621 Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 2^c*5^(3d) with a,b,c,d nonnegative integers.

1, 2, 2, 3, 3, 2, 3, 5, 2, 4, 4, 3, 3, 5, 3, 3, 6, 4, 4, 5, 2, 6, 5, 4, 4, 5, 2, 4, 6, 3, 4, 8, 5, 3, 5, 4, 5, 8, 5, 5, 4, 3, 5, 7, 4, 5, 8, 4, 2, 8, 2, 6, 7, 4, 3, 4, 6, 5, 8, 4, 4, 6, 5, 5, 5, 5, 6, 8, 4, 6, 7, 4, 6, 10, 4, 4, 7, 5, 2, 10, 4, 7, 7, 4, 8, 4, 4, 7, 8, 2, 4, 9, 5, 5, 9, 5, 5, 7, 5, 5

Offset

1,2

Comments

Conjecture: a(n) > 0 for all n > 0. Equivalently, any positive integer n can be written as a^2 + b*(b+1) + 2^c*5^(3d) with a,b,c,d nonnegative integers.

This was motivated by A308584, and we verified a(n) > 0 for all n = 1..2*10^8. Then, on the author's request, Giovanni Resta verified the above conjecture for n up to 10^10. G. Resta also noted that 7272774228 cannot be written as a*(a+1)/2 + b*(b+1)/2 + 2^c*250^d with a,b,c,d nonnegative integers.

See also A308566, A308584 and A308623 for similar conjectures.

Links

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

Example

a(1) = 1 with 1 = 0*1/2 + 0*1/2 + 2^0*5^(3*0).
a(78210) = 1 with 78210 = 85*86/2 + 385*386/2 + 2*5^3.

Mathematica

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={}; Do[r=0; Do[If[TQ[n-5^(3k)*2^m-x(x+1)/2], r=r+1], {k, 0, Log[5, n]/3}, {m, 0, Log[2, n/5^(3k)]}, {x, 0, (Sqrt[4(n-5^(3k)*2^m)+1]-1)/2}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000079, A000217, A000351, A308566, A308584, A308623.

Sequence in context: A039645 A317993 A048687 * A308623 A115074 A234399

Adjacent sequences: A308618 A308619 A308620 * A308622 A308623 A308624

Keyword

nonn

Author

Zhi-Wei Sun, Jun 11 2019

Status

approved