A325248 Heinz number of the omega-sequence of n.

1, 2, 2, 6, 2, 18, 2, 10, 6, 18, 2, 90, 2, 18, 18, 14, 2, 90, 2, 90, 18, 18, 2, 126, 6, 18, 10, 90, 2, 50, 2, 22, 18, 18, 18, 42, 2, 18, 18, 126, 2, 50, 2, 90, 90, 18, 2, 198, 6, 90, 18, 90, 2, 126, 18, 126, 18, 18, 2, 630, 2, 18, 90, 26, 18, 50, 2, 90, 18, 50

Offset: 1

Author

Gus Wiseman, Apr 16 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).

Examples

			The omega-sequence of 180 is (5,3,2,2,1) with Heinz number 990, so a(180) = 990.
		

Crossrefs

Positions of squarefree terms are A325247.

Positions of normal numbers (A055932) are A325251.

First positions of each distinct term are A325238.

Cf. A056239, A070175, A112798, A118914, A181819, A181821, A323023, A325249, A325275, 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).

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,100}]

Formula

A001222(a(n)) = A323014(n).

A061395(a(n)) = A001222(n).

A304465(n) = A055396(a(n)/2).

A325249(n) = A056239(a(n)).

a(n!) = A325275(n).