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A325264
Numbers whose omega-sequence sums to 7.
4
30, 36, 42, 64, 66, 70, 78, 100, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 196, 222, 225, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426
OFFSET
1,1
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).
EXAMPLE
The sequence of terms together with their prime indices and omega-sequences begins:
30: {1,2,3} (3,3,1)
36: {1,1,2,2} (4,2,1)
42: {1,2,4} (3,3,1)
64: {1,1,1,1,1,1} (6,1)
66: {1,2,5} (3,3,1)
70: {1,3,4} (3,3,1)
78: {1,2,6} (3,3,1)
100: {1,1,3,3} (4,2,1)
102: {1,2,7} (3,3,1)
105: {2,3,4} (3,3,1)
110: {1,3,5} (3,3,1)
114: {1,2,8} (3,3,1)
130: {1,3,6} (3,3,1)
138: {1,2,9} (3,3,1)
154: {1,4,5} (3,3,1)
165: {2,3,5} (3,3,1)
170: {1,3,7} (3,3,1)
174: {1,2,10} (3,3,1)
182: {1,4,6} (3,3,1)
186: {1,2,11} (3,3,1)
190: {1,3,8} (3,3,1)
195: {2,3,6} (3,3,1)
196: {1,1,4,4} (4,2,1)
MATHEMATICA
omseq[n_Integer]:=If[n<=1, {}, Total/@NestWhileList[Sort[Length/@Split[#]]&, Sort[Last/@FactorInteger[n]], Total[#]>1&]];
Select[Range[100], Total[omseq[#]]==7&]
CROSSREFS
Positions of 7's in A325249.
Numbers with omega-sequence summing to m: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7).
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), A325249 (sum).
Sequence in context: A214408 A350353 A357874 * A345382 A227680 A109426
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 18 2019
STATUS
approved