A236473 Number of partitions into multiplicatively perfect numbers, cf. A007422.

1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 8, 10, 10, 12, 12, 15, 17, 21, 22, 26, 27, 32, 35, 41, 44, 52, 55, 63, 68, 78, 85, 98, 105, 119, 128, 144, 156, 177, 191, 214, 231, 257, 277, 310, 335, 372, 402, 444, 478, 529, 571, 630, 681, 747, 804, 883, 951

Offset: 0

Author

Reinhard Zumkeller, Jan 26 2014

Keywords

• nonn

Examples

			a(10) = #{10, 8+1+1, 6+1+1+1+1, 10x1} = 4;
a(11) = #{10+1, 8+1+1+1, 6+1+1+1+1+1, 11x1} = 4;
a(12) = #{10+1+1, 8+1+1+1+1, 6+6, 6+6x1, 12x1} = 5;
a(13) = #{10+1+1+1, 8+1+1+1+1+1, 6+6+1, 6+7x1, 13x1} = 5;
a(14) = #{14, 10+1+1+1+1, 8+6, 8+6x1, 6+6+1+1, 6+8x1, 14x1} = 7;
a(15) = #{15, 14+1, 10+1+1+1+1+1, 8+6+1, 8+7x1, 6+6+1+1+1, 6+9x1, 15x1} = 8;
a(16) = #{15+1, 14+1+1, 10+6, 10+6x1, 8+8, 8+6+1+1, 8+8x1, 6+6+1+1+1+1, 6+10x1, 16x1} = 10.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Programs

Haskell

a236473 = p a007422_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*(add(
      `if`(tau(d)=4, d, 0), d=divisors(j))+1), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 23 2017

Mathematica

a[n_] := a[n] = If[n == 0, 1, Sum[a[n-j]*(Sum[If[DivisorSigma[0, d] == 4, d, 0], {d, Divisors[j]}] + 1), {j, 1, n}]/n];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 12 2017, after Alois P. Heinz *)