A116900 Number of partitions of n into at least two parts such that the product of largest and smallest part is equal to n.

0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 2, 0, 2, 1, 5, 0, 7, 0, 7, 4, 8, 0, 21, 1, 14, 13, 25, 0, 43, 0, 44, 31, 41, 2, 121, 0, 66, 73, 126, 0, 215, 0, 193, 179, 165, 0, 554, 1, 285, 346, 491, 0, 890, 65, 772, 704, 574, 0, 2330, 0, 847, 1392, 1828, 254, 3212, 0, 2754, 2649, 2282, 0, 7907

Offset

0,13

Comments

Clearly a(p)=0, when p is prime.

Number of partitions p of n such that mean(p) = multiplicity(max(p)). For example, a(12) counts these two partitions: 441111, 332211. See the Mathematica program at A240200 for a count of partitions defined in this manner, along with related sequences. - Clark Kimberling, Apr 03 2014

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

a(n) = coefficient of x^n in expansion of Sum_{d|n} x^(d+n/d)/Product(1-x^k, k=d..n/d). - Vladeta Jovovic, Nov 24 2008

Example

a(21) = 4 since property holds for 4 partitions of 21: (7,7,4,3), (7,6,5,3), (7,5,3,3,3), (7,4,4,3,3).

Maple

b:= proc(n, i, m) option remember; `if`(n=0, 1,
      `if`(i<m, 0, add(b(n-i*j, i-1, m), j=0..n/i)))
    end:
a:= n-> add(b(n-d-n/d, d, n/d), d=select(x->
        is(x>=sqrt(n)), numtheory[divisors](n))):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 03 2014

Mathematica

f[n_] := Length@ Select[ IntegerPartitions@n, (Length@ # > 1 && Last@# First@# == n) &]; Array[f, 72] (* Robert G. Wilson v, Mar 15 2006 *)
b[n_, i_, m_] := b[n, i, m] = If[n==0, 1, If[i<m, 0, Sum[b[n-i*j, i-1, m], {j, 0, n/i}]]]; a[n_] := Sum[b[n-d-n/d, d, n/d], {d, Select[Divisors[n], # >= Sqrt[n]&]}]; a[0] = 0; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Sep 12 2015, after Alois P. Heinz *)

Crossrefs

Cf. A000041, A116901, A116902, A240200.

Sequence in context: A081082 A049785 A036997 * A254372 A354158 A363580

Adjacent sequences: A116897 A116898 A116899 * A116901 A116902 A116903

Keyword

nonn

Author

Giovanni Resta, Mar 14 2006

Extensions

More terms from Robert G. Wilson v, Mar 15 2006

Status

approved