A168657 Number of partitions of n such that the number of parts is divisible by the smallest part.

1, 1, 2, 4, 6, 8, 12, 17, 25, 34, 48, 64, 87, 114, 151, 198, 258, 332, 428, 546, 695, 879, 1108, 1388, 1737, 2159, 2680, 3312, 4082, 5009, 6138, 7492, 9126, 11081, 13429, 16228, 19575, 23547, 28277, 33879, 40520, 48354, 57615, 68509, 81337, 96388, 114055

Offset

1,3

Links

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

Formula

G.f.: Sum_{n>=1} Sum_{d|n} x^(n*d)/Product_{k=1..n-1}(1-x^k).

G.f.: Sum_{i>=1} Sum_{j>=1} x^(i*j^2)/Product_{k=1..i*j-1} (1-x^k). - Seiichi Manyama, Jan 21 2022

Maple

b:= proc(n, i, t) option remember;
      `if`(n<1, 0, `if`(i=1, 1, `if`(i<1, 0,
      `if`(irem(n, i)=0 and irem(t+n/i, i)=0, 1, 0)+
            add(b(n-i*j, i-1, t+j), j=0..n/i))))
    end:
a:= n-> b(n, n, 0):
seq(a(n), n=1..60);  # Alois P. Heinz, May 24 2012

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n<1, 0, If[i==1, 1, If[i<1, 0, If [Mod[n, i]==0 && Mod[t+n/i, i]==0, 1, 0] + Sum[b[n-i*j, i-1, t+j], {j, 0, n/i}]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jul 01 2015, after Alois P. Heinz *)

Prog

(PARI) my(N=66, x='x+O('x^N)); Vec(sum(i=1, N, sum(j=1, sqrtint(N\i), x^(i*j^2)/prod(k=1, i*j-1, 1-x^k)))) \\ Seiichi Manyama, Jan 21 2022

Crossrefs

Cf. A079501, A168655, A168656.

Sequence in context: A181821 A241882 A238626 * A240498 A081954 A096903

Adjacent sequences: A168654 A168655 A168656 * A168658 A168659 A168660

Keyword

easy,nonn

Author

Vladeta Jovovic, Dec 02 2009

Status

approved