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 A236473 Number of partitions into multiplicatively perfect numbers, cf. A007422. 5

%I

%S 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 41,44,52,55,63,68,78,85,98,105,119,128,144,156,177,191,214,231,257,

%U 277,310,335,372,402,444,478,529,571,630,681,747,804,883,951

%N Number of partitions into multiplicatively perfect numbers, cf. A007422.

%H Alois P. Heinz, <a href="/A236473/b236473.txt">Table of n, a(n) for n = 0..10000</a>

%e a(10) = #{10, 8+1+1, 6+1+1+1+1, 10x1} = 4;

%e a(11) = #{10+1, 8+1+1+1, 6+1+1+1+1+1, 11x1} = 4;

%e a(12) = #{10+1+1, 8+1+1+1+1, 6+6, 6+6x1, 12x1} = 5;

%e a(13) = #{10+1+1+1, 8+1+1+1+1+1, 6+6+1, 6+7x1, 13x1} = 5;

%e a(14) = #{14, 10+1+1+1+1, 8+6, 8+6x1, 6+6+1+1, 6+8x1, 14x1} = 7;

%e 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 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 with(numtheory):

%p `if`(tau(d)=4, d, 0), d=divisors(j))+1), j=1..n)/n)

%p end:

%p seq(a(n), n=0..100); # _Alois P. Heinz_, Mar 23 2017

%t 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 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Apr 12 2017, after _Alois P. Heinz_ *)

%o a236473 = p a007422_list where

%o p _ 0 = 1

%o p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

%K nonn

%O 0,7

%A _Reinhard Zumkeller_, Jan 26 2014

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Last modified August 1 10:37 EDT 2021. Contains 346385 sequences. (Running on oeis4.)