A308623 Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 2^c*10^(2d) 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, 6

Offset

1,2

Comments

Conjecture: a(n) > 0 for all n > 0. Equivalently, any positive integer n can be written as a*(a+1)/2 + b*(b+1)/2 + 2^c*10^(2d) with a,b,c,d nonnegative integers.

This was motivated by A308566, 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 729546026 cannot be written as a*(a+1)/2 + b*(b+1)/2 + 2^c*3^d with a,b,c,d nonnegative integers.

See also A308566, A308594 and A308621 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*10^(2*0).
a(10107) = 1 with 10107 = 82*83/2 + 96*97/2 + 2^11*10^(2*0).

Mathematica

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={}; Do[r=0; Do[If[TQ[n-10^(2k)*2^m-x(x+1)/2], r=r+1], {k, 0, Log[10, n]/2}, {m, 0, Log[2, n/10^(2k)]}, {x, 0, (Sqrt[4(n-10^(2k)*2^m)+1]-1)/2}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000079, A000217, A011557, A308566, A308584, A308621.

Sequence in context: A317993 A048687 A308621 * A115074 A234399 A214749

Adjacent sequences: A308620 A308621 A308622 * A308624 A308625 A308626

Keyword

nonn

Author

Zhi-Wei Sun, Jun 11 2019

Status

approved