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A331740 Number of prime factors in A225546(n), counted with multiplicity.

%I A331740 #18 Oct 06 2021 19:41:44
%S A331740 0,1,2,1,4,3,8,2,2,5,16,3,32,9,6,1,64,3,128,5,10,17,256,4,4,33,4,9,
%T A331740 512,7,1024,2,18,65,12,3,2048,129,34,6,4096,11,8192,17,6,257,16384,3,
%U A331740 8,5,66,33,32768,5,20,10,130,513,65536,7,131072,1025,10,2,36,19,262144,65,258,13,524288,4,1048576,2049,6
%N A331740 Number of prime factors in A225546(n), counted with multiplicity.
%H A331740 Antti Karttunen, <a href="/A331740/b331740.txt">Table of n, a(n) for n = 1..16384</a>
%H A331740 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A331740 Additive with a(p^e) = A000120(e) * 2^(PrimePi(p)-1), where PrimePi(n) = A000720(n).
%F A331740 a(n) = A001222(A225546(n)).
%F A331740 A331591(n) <= a(n) <= A048675(n).
%F A331740 From _Peter Munn_, Sep 11 2021: (Start)
%F A331740 a(A001146(m)) = 1.
%F A331740 a(A331590(m, k)) = a(m) + a(k).
%F A331740 For squarefree k, a(k*m^2) = a(k) + a(m) = A048675(k) + a(m).
%F A331740 (End)
%t A331740 Array[If[# == 1, 0, PrimeOmega@ Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]] &, 75] (* _Michael De Vlieger_, Feb 08 2020 *)
%o A331740 (PARI) A331740(n) = if(1==n,0,my(f=factor(n)); sum(i=1,#f~,hammingweight(f[i,2])*(2^(primepi(f[i,1])-1))));
%Y A331740 Cf. A000120, A000720, A001222, A048675, A225546, A331590.
%Y A331740 Cf. also A331309, A331591.
%Y A331740 Positions of 1's: A001146.
%K A331740 nonn
%O A331740 1,3
%A A331740 _Antti Karttunen_, Feb 05 2020