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%I #4 Apr 18 2019 16:55:52
%S 1,1,2,18,126,990,850,11970,19530,25830,4606,73458,92862,116298,43134,
%T 229086,275418,366894,440946,515394,568062,613206,769158,963378,
%U 1060254,1135602,6108570,6431490,6915870,8923590,9398610,10191870,11352510,3139866,16458210
%N Heinz number 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).
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%t omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
%t Table[Times@@Prime/@omseq[n!],{n,30}]
%Y A001222(a(n)) = A325272.
%Y A055396(a(n)/2) = A325273.
%Y A056239(a(n)) = A325274.
%Y Row n of A325276 is row a(n) of A112798.
%Y Cf. A000142, A006939, A056239, A112798, A303555, A323023, A325238, 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 0,3
%A _Gus Wiseman_, Apr 18 2019
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