%I #10 Aug 22 2019 09:53:58

%S 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 53,59,61,64,67,71,73,79,81,83,89,97,100,101,103,107,109,113,121,125,

%U 127,128,131,137,139,149,151,157,163,167,169,173,179,181,191,193

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

%C First differs from A323306 in having 216.

%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).

%C 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 The sequence of terms together with their prime indices begins:

%e 1: {}

%e 2: {1}

%e 3: {2}

%e 4: {1,1}

%e 5: {3}

%e 7: {4}

%e 8: {1,1,1}

%e 9: {2,2}

%e 11: {5}

%e 13: {6}

%e 16: {1,1,1,1}

%e 17: {7}

%e 19: {8}

%e 23: {9}

%e 25: {3,3}

%e 27: {2,2,2}

%e 29: {10}

%e 31: {11}

%e 32: {1,1,1,1,1}

%e 36: {1,1,2,2}

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

%t Select[Range[100],UnsameQ@@omseq[#]&]

%Y Positions of squarefree numbers in A325248.

%Y Cf. A056239, A112798, A118914, A181819, A323023, A325249, A325250, A325251, A325277.

%Y 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 nonn

%O 1,2

%A _Gus Wiseman_, Apr 16 2019