A274046 a(n) is the smallest positive integer which can be represented as the sum of distinct positive triangular numbers in exactly n ways, or 0 if no such integer exists.

1, 10, 25, 31, 49, 46, 55, 67, 70, 76, 82, 117, 102, 91, 97, 107, 101, 135, 110, 112, 116, 115, 119, 128, 0, 131, 133, 130, 148, 145, 136, 0, 137, 149, 154, 146, 0, 169, 152, 157, 155, 168, 171, 158, 174, 161, 0, 183, 184, 167, 0, 0, 173, 0, 175, 181, 190

Offset: 1

Author

Phil Scovis, Jun 07 2016

Keywords

• nonn

Comments

46 is the smallest number that can be expressed as the sum of distinct triangular numbers in five ways, but 49 is the smallest that can be so expressed in exactly five ways. There are further examples of this phenomenon.

Examples

			25 = 1 + 3 + 6 + 15 = 10 + 15 = 1 + 3 + 21. This is the smallest number that can be written as the sum of distinct triangular numbers in three different ways. So a(3)=25.
The first null values of a(n) occur for n = 25, 32, 37, 47, 51, 52, 54, 61,... - _Giovanni Resta_, Jun 08 2016
		

Crossrefs

Cf. A007294, A060773, A024940, A064816.

Programs

Mathematica

nT[n_, m_: 0] := nT[n,m] = If[n == 0, 1, Block[{t, i=m+1, s=0}, While[(t = i*(i+1)/2) <= n, s += nT[n-t, i]; i++]; s]]; a[n_] := Block[{k=0, t}, While[(t = nT[++k]) != n && t < Max[2*n, 30]]; If[t == n, k, 0]]; Array[a, 57] (* Giovanni Resta, Jun 08 2016 *)

Extensions

a(15)-a(20) from Tom Edgar, Jun 08 2016

a(21)-a(57) from Giovanni Resta, Jun 08 2016