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A342621 Sum of the partition number of the prime factors of n with multiplicity.

%I A342621 #25 Apr 09 2021 19:51:39
%S A342621 0,2,3,4,7,5,15,6,6,9,56,7,101,17,10,8,297,8,490,11,18,58,1255,9,14,
%T A342621 103,9,19,4565,12,6842,10,59,299,22,10,21637,492,104,13,44583,20,
%U A342621 63261,60,13,1257,124754,11,30,16,300,105,329931,11,63,21,493,4567,831820
%N A342621 Sum of the partition number of the prime factors of n with multiplicity.
%H A342621 Alois P. Heinz, <a href="/A342621/b342621.txt">Table of n, a(n) for n = 1..10000</a> (first 3000 terms from Eric Desbiaux)
%F A342621 a(A003586(n)) - A001414(A003586(n)) = 0.
%F A342621 a(A006899(n)) * A008480(A006899(n)) - A001414(A006899(n)) = 0.
%F A342621 a(n) = Sum_{k=1..A001222(n)} A000041(A027746(n,k)). - _Alois P. Heinz_, Apr 09 2021
%e A342621 For n = 408 = 2^3*3*17, a(408) = 3 * A000041(2) + A000041(3) + A000041(17) = 3*2 + 3 + 297 = 306.
%p A342621 a:= n-> add(combinat[numbpart](i[1])*i[2], i=ifactors(n)[2]):
%p A342621 seq(a(n), n=1..70);  # _Alois P. Heinz_, Mar 17 2021
%t A342621 {0}~Join~Array[Total@ Map[#2 PartitionsP[#1] & @@ # &, FactorInteger[#]] &, 58, 2] (* _Michael De Vlieger_, Mar 17 2021 *)
%o A342621 (Sage)
%o A342621 def a(n):
%o A342621     return sum([Partitions(primefactor).cardinality() for (primefactor,exponent) in factor(n) for _ in range(exponent)])
%o A342621 [a(n) for n in (1..100)]
%o A342621 (PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,2]*numbpart(f[k,1])); \\ _Michel Marcus_, Mar 17 2021
%Y A342621 Cf. A000041, A027746, A058698, A001222.
%K A342621 nonn
%O A342621 1,2
%A A342621 _Eric Desbiaux_, Mar 16 2021