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

%I A236473 #9 Apr 12 2017 04:43:19
%S A236473 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,
%T A236473 41,44,52,55,63,68,78,85,98,105,119,128,144,156,177,191,214,231,257,
%U A236473 277,310,335,372,402,444,478,529,571,630,681,747,804,883,951
%N A236473 Number of partitions into multiplicatively perfect numbers, cf. A007422.
%H A236473 Alois P. Heinz, <a href="/A236473/b236473.txt">Table of n, a(n) for n = 0..10000</a>
%e A236473 a(10) = #{10, 8+1+1, 6+1+1+1+1, 10x1} = 4;
%e A236473 a(11) = #{10+1, 8+1+1+1, 6+1+1+1+1+1, 11x1} = 4;
%e A236473 a(12) = #{10+1+1, 8+1+1+1+1, 6+6, 6+6x1, 12x1} = 5;
%e A236473 a(13) = #{10+1+1+1, 8+1+1+1+1+1, 6+6+1, 6+7x1, 13x1} = 5;
%e A236473 a(14) = #{14, 10+1+1+1+1, 8+6, 8+6x1, 6+6+1+1, 6+8x1, 14x1} = 7;
%e A236473 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;
%e A236473 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.
%p A236473 with(numtheory):
%p A236473 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*(add(
%p A236473       `if`(tau(d)=4, d, 0), d=divisors(j))+1), j=1..n)/n)
%p A236473     end:
%p A236473 seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 23 2017
%t A236473 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];
%t A236473 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Apr 12 2017, after _Alois P. Heinz_ *)
%o A236473 (Haskell)
%o A236473 a236473 = p a007422_list where
%o A236473    p _          0 = 1
%o A236473    p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%K A236473 nonn
%O A236473 0,7
%A A236473 _Reinhard Zumkeller_, Jan 26 2014