cp's OEIS Frontend

A325247 Numbers whose omega-sequence is strict (no repeated parts).

%I A325247 #10 Aug 22 2019 09:53:58
%S A325247 1,2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,36,37,41,43,47,49,
%T A325247 53,59,61,64,67,71,73,79,81,83,89,97,100,101,103,107,109,113,121,125,
%U A325247 127,128,131,137,139,149,151,157,163,167,169,173,179,181,191,193
%N A325247 Numbers whose omega-sequence is strict (no repeated parts).
%C A325247 First differs from A323306 in having 216.
%C A325247 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).
%C A325247 Also Heinz numbers of integer partitions of whose omega-sequence is strict (counted by A325250). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e A325247 The sequence of terms together with their prime indices begins:
%e A325247      1: {}
%e A325247      2: {1}
%e A325247      3: {2}
%e A325247      4: {1,1}
%e A325247      5: {3}
%e A325247      7: {4}
%e A325247      8: {1,1,1}
%e A325247      9: {2,2}
%e A325247     11: {5}
%e A325247     13: {6}
%e A325247     16: {1,1,1,1}
%e A325247     17: {7}
%e A325247     19: {8}
%e A325247     23: {9}
%e A325247     25: {3,3}
%e A325247     27: {2,2,2}
%e A325247     29: {10}
%e A325247     31: {11}
%e A325247     32: {1,1,1,1,1}
%e A325247     36: {1,1,2,2}
%t A325247 omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#1]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
%t A325247 Select[Range[100],UnsameQ@@omseq[#]&]
%Y A325247 Positions of squarefree numbers in A325248.
%Y A325247 Cf. A056239, A112798, A118914, A181819, A323023, A325249, A325250, A325251, A325277.
%Y A325247 Omega-sequence statistics: A001221 (second omega), A001222 (first omega), A071625 (third omega), A304465 (second-to-last omega), A182850 or A323014 (depth), A323022 (fourth omega), A325248 (Heinz number).
%K A325247 nonn
%O A325247 1,2
%A A325247 _Gus Wiseman_, Apr 16 2019