A325416 - OEIS

A325416

Least k such that the omega-sequence of k sums to n, and 0 if none exists.

1, 2, 0, 4, 8, 6, 32, 30, 12, 24, 48, 96, 60, 120, 240, 480, 960, 1920, 3840, 2520, 5040, 10080, 20160, 40320, 80640

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OFFSET

0,2

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) with sum 13.

LINKS

Table of n, a(n) for n=0..24.

EXAMPLE

The sequence of terms together with their omega-sequences (n = 2 term not shown) begins:

2: 1

4: 2 1

8: 3 1

6: 2 2 1

32: 5 1

30: 3 3 1

12: 3 2 2 1

24: 4 2 2 1

48: 5 2 2 1

96: 6 2 2 1

60: 4 3 2 2 1

120: 5 3 2 2 1

240: 6 3 2 2 1

480: 7 3 2 2 1

960: 8 3 2 2 1

1920: 9 3 2 2 1

3840: 10 3 2 2 1

2520: 7 4 3 2 2 1

5040: 8 4 3 2 2 1

MATHEMATICA

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

da=Table[Total[omseq[n]], {n, 10000}];

Table[If[!MemberQ[da, k], 0, Position[da, k][[1, 1]]], {k, 0, Max@@da}]

CROSSREFS

Cf. A056239, A181819, A181821, A304465, A307734, A323023, A325238, A325277, A325280, A325413, A325415.

Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (frequency depth), A325248 (Heinz number), A325249 (sum).

Sequence in context: A021087 A120558 A354042 * A372492 A120554 A120710

Adjacent sequences: A325413 A325414 A325415 * A325417 A325418 A325419

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 25 2019

STATUS

approved