A325275 Heinz number of the omega-sequence of n!.

1, 1, 2, 18, 126, 990, 850, 11970, 19530, 25830, 4606, 73458, 92862, 116298, 43134, 229086, 275418, 366894, 440946, 515394, 568062, 613206, 769158, 963378, 1060254, 1135602, 6108570, 6431490, 6915870, 8923590, 9398610, 10191870, 11352510, 3139866, 16458210

Offset: 0

Author

Gus Wiseman, Apr 18 2019

Keywords

• nonn

Comments

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).

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Crossrefs

A001222(a(n)) = A325272.

A055396(a(n)/2) = A325273.

A056239(a(n)) = A325274.

Row n of A325276 is row a(n) of A112798.

Cf. A000142, A006939, A056239, A112798, A303555, A323023, A325238, A325277.

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).

Programs

Mathematica

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Table[Times@@Prime/@omseq[n!],{n,30}]