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%I #4 Apr 18 2019 16:55:45
%S 0,0,1,5,9,13,14,20,23,25,24,30,33,35,35,40,44,46,49,51,54,56,59,61,
%T 65,67,72,75,78,80,83,85,90,90,95,97,101,103,105,106,110,112,115,117,
%U 122,125,127,129,134,136,139,140,143,145,149,153,157,159,160,162
%N Sum of the omega-sequence of n!.
%C We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1), with sum 13.
%t omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
%t Table[Total[omseq[n!]],{n,0,100}]
%Y a(n) = A056239(A325275(n)).
%Y Cf. A000142, A006939, A303555, A323023, A325238, A325272, A325273, A325276, A325277.
%Y Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).
%K nonn
%O 0,4
%A _Gus Wiseman_, Apr 18 2019