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A325416
Least k such that the omega-sequence of k sums to n, and 0 if none exists.
4
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
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.
EXAMPLE
The sequence of terms together with their omega-sequences (n = 2 term not shown) begins:
1:
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
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 25 2019
STATUS
approved