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Numbers whose omega-sequence sums to 7.
4

%I #5 Apr 18 2019 16:54:43

%S 30,36,42,64,66,70,78,100,102,105,110,114,130,138,154,165,170,174,182,

%T 186,190,195,196,222,225,230,231,238,246,255,258,266,273,282,285,286,

%U 290,310,318,322,345,354,357,366,370,374,385,399,402,406,410,418,426

%N Numbers whose omega-sequence sums to 7.

%C 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).

%e The sequence of terms together with their prime indices and omega-sequences begins:

%e 30: {1,2,3} (3,3,1)

%e 36: {1,1,2,2} (4,2,1)

%e 42: {1,2,4} (3,3,1)

%e 64: {1,1,1,1,1,1} (6,1)

%e 66: {1,2,5} (3,3,1)

%e 70: {1,3,4} (3,3,1)

%e 78: {1,2,6} (3,3,1)

%e 100: {1,1,3,3} (4,2,1)

%e 102: {1,2,7} (3,3,1)

%e 105: {2,3,4} (3,3,1)

%e 110: {1,3,5} (3,3,1)

%e 114: {1,2,8} (3,3,1)

%e 130: {1,3,6} (3,3,1)

%e 138: {1,2,9} (3,3,1)

%e 154: {1,4,5} (3,3,1)

%e 165: {2,3,5} (3,3,1)

%e 170: {1,3,7} (3,3,1)

%e 174: {1,2,10} (3,3,1)

%e 182: {1,4,6} (3,3,1)

%e 186: {1,2,11} (3,3,1)

%e 190: {1,3,8} (3,3,1)

%e 195: {2,3,6} (3,3,1)

%e 196: {1,1,4,4} (4,2,1)

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

%t Select[Range[100],Total[omseq[#]]==7&]

%Y Positions of 7's in A325249.

%Y 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).

%Y Cf. A056239, A060687, A062770, A118914, A130091, A303555, A323023, A325238, A325265, A325277.

%Y 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).

%K nonn

%O 1,1

%A _Gus Wiseman_, Apr 18 2019