cp's OEIS Frontend

A325264 Numbers whose omega-sequence sums to 7.

%I A325264 #5 Apr 18 2019 16:54:43
%S A325264 30,36,42,64,66,70,78,100,102,105,110,114,130,138,154,165,170,174,182,
%T A325264 186,190,195,196,222,225,230,231,238,246,255,258,266,273,282,285,286,
%U A325264 290,310,318,322,345,354,357,366,370,374,385,399,402,406,410,418,426
%N A325264 Numbers whose omega-sequence sums to 7.
%C A325264 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 A325264 The sequence of terms together with their prime indices and omega-sequences begins:
%e A325264    30: {1,2,3} (3,3,1)
%e A325264    36: {1,1,2,2} (4,2,1)
%e A325264    42: {1,2,4} (3,3,1)
%e A325264    64: {1,1,1,1,1,1} (6,1)
%e A325264    66: {1,2,5} (3,3,1)
%e A325264    70: {1,3,4} (3,3,1)
%e A325264    78: {1,2,6} (3,3,1)
%e A325264   100: {1,1,3,3} (4,2,1)
%e A325264   102: {1,2,7} (3,3,1)
%e A325264   105: {2,3,4} (3,3,1)
%e A325264   110: {1,3,5} (3,3,1)
%e A325264   114: {1,2,8} (3,3,1)
%e A325264   130: {1,3,6} (3,3,1)
%e A325264   138: {1,2,9} (3,3,1)
%e A325264   154: {1,4,5} (3,3,1)
%e A325264   165: {2,3,5} (3,3,1)
%e A325264   170: {1,3,7} (3,3,1)
%e A325264   174: {1,2,10} (3,3,1)
%e A325264   182: {1,4,6} (3,3,1)
%e A325264   186: {1,2,11} (3,3,1)
%e A325264   190: {1,3,8} (3,3,1)
%e A325264   195: {2,3,6} (3,3,1)
%e A325264   196: {1,1,4,4} (4,2,1)
%t A325264 omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
%t A325264 Select[Range[100],Total[omseq[#]]==7&]
%Y A325264 Positions of 7's in A325249.
%Y A325264 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 A325264 Cf. A056239, A060687, A062770, A118914, A130091, A303555, A323023, A325238, A325265, A325277.
%Y A325264 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 A325264 nonn
%O A325264 1,1
%A A325264 _Gus Wiseman_, Apr 18 2019