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

1,2

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.

Links

Table of n, a(n) for n=1..57.

Example

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

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 *)

Crossrefs

Cf. A007294, A060773, A024940, A064816.

Sequence in context: A048195 A133634 A174051 * A014090 A345651 A154057

Adjacent sequences: A274043 A274044 A274045 * A274047 A274048 A274049

Keyword

nonn

Author

Phil Scovis, Jun 07 2016

Extensions

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

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

Status

approved