A117086 Number of partitions of n such that the largest part is a multiple of the smallest part.

1, 2, 3, 5, 6, 11, 12, 20, 26, 37, 45, 71, 84, 117, 152, 203, 253, 342, 421, 556, 694, 884, 1096, 1409, 1729, 2168, 2672, 3327, 4061, 5039, 6114, 7514, 9110, 11098, 13400, 16275, 19537, 23575, 28245, 33929, 40465, 48424, 57552, 68569, 81296, 96449

Offset

1,2

Comments

Also number of partitions of n such that the number of parts is a multiple of the multiplicity of the largest part. Example: a(7)=12 because from the 15 (=A000041(7)) partitions of 7 only [3,3,1], [2,2,2,1] and [2,2,1,1,1] do not qualify (3,4,5 are not multiples of 2,3,2, respectively). - Emeric Deutsch, Apr 21 2006

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

G.f.: Sum_{L>=0} Sum_{k>=1} (x^((L+1)*k) / Product_{i=k..L*k} (1 - x^i)).

Example

a(7)=12 because from the 15 (=A000041(7)) partitions of 7 only [5,2],[4,3] and [3,2,2] do not qualify.

Maple

f:=add(add(x^((l+1)*k)/mul(1-x^i, i=k..l*k), k=1..51), l=0..51): s:=series(f, x, 51):for m from 1 to 50 do c:=coeff(s, x, m): printf(`%d, `, c); od: # (Jovovic) - Emeric Deutsch, Apr 21 2006

Mathematica

Table[Count[IntegerPartitions[n], _?(Divisible[First[#], Last[#]]&)], {n, 50}] (* Harvey P. Dale, Mar 04 2012 *)

Crossrefs

Cf. A118096.

Cf. A000041.

Sequence in context: A365311 A083710 A127524 * A344551 A081026 A137808

Adjacent sequences: A117083 A117084 A117085 * A117087 A117088 A117089

Keyword

easy,nonn

Author

Vladeta Jovovic, Apr 17 2006

Extensions

More terms from Emeric Deutsch, Apr 21 2006

Status

approved