

A325248


Heinz number of the omegasequence of n.


26



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 omegasequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the kth term is Omega(red^{k1}(n)), where Omega = A001222 and red^{k} is the kth 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 omegasequence 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 omegasequence 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.
Omegasequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (secondtolast 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



