A325248 - OEIS

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

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OFFSET

1,2

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).

LINKS

Table of n, a(n) for n=1..70.

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).

EXAMPLE

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

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}]

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).

Sequence in context: A057562 A102628 A211776 * A036655 A319356 A098792

Adjacent sequences: A325245 A325246 A325247 * A325249 A325250 A325251

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 16 2019

STATUS

approved