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%I #12 Aug 22 2019 09:55:07
%S 6,10,12,14,15,18,20,21,22,24,26,28,30,33,34,35,38,39,40,42,44,45,46,
%T 48,50,51,52,54,55,56,57,58,60,62,63,65,66,68,69,70,72,74,75,76,77,78,
%U 80,82,84,85,86,87,88,90,91,92,93,94,95,96,98,99,102,104,105
%N Numbers whose omega-sequence has repeated parts.
%C First differs from A323304 in lacking 216. First differs from A106543 in having 144.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose omega-sequence has repeated parts. The enumeration of these partitions by sum is given by A325285.
%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), which has repeated parts, so 180 is in the sequence.
%e The sequence of terms together with their omega-sequences begins:
%e 6: 2 2 1 51: 2 2 1 86: 2 2 1 119: 2 2 1
%e 10: 2 2 1 52: 3 2 2 1 87: 2 2 1 120: 5 3 2 2 1
%e 12: 3 2 2 1 54: 4 2 2 1 88: 4 2 2 1 122: 2 2 1
%e 14: 2 2 1 55: 2 2 1 90: 4 3 2 2 1 123: 2 2 1
%e 15: 2 2 1 56: 4 2 2 1 91: 2 2 1 124: 3 2 2 1
%e 18: 3 2 2 1 57: 2 2 1 92: 3 2 2 1 126: 4 3 2 2 1
%e 20: 3 2 2 1 58: 2 2 1 93: 2 2 1 129: 2 2 1
%e 21: 2 2 1 60: 4 3 2 2 1 94: 2 2 1 130: 3 3 1
%e 22: 2 2 1 62: 2 2 1 95: 2 2 1 132: 4 3 2 2 1
%e 24: 4 2 2 1 63: 3 2 2 1 96: 6 2 2 1 133: 2 2 1
%e 26: 2 2 1 65: 2 2 1 98: 3 2 2 1 134: 2 2 1
%e 28: 3 2 2 1 66: 3 3 1 99: 3 2 2 1 135: 4 2 2 1
%e 30: 3 3 1 68: 3 2 2 1 102: 3 3 1 136: 4 2 2 1
%e 33: 2 2 1 69: 2 2 1 104: 4 2 2 1 138: 3 3 1
%e 34: 2 2 1 70: 3 3 1 105: 3 3 1 140: 4 3 2 2 1
%e 35: 2 2 1 72: 5 2 2 1 106: 2 2 1 141: 2 2 1
%e 38: 2 2 1 74: 2 2 1 108: 5 2 2 1 142: 2 2 1
%e 39: 2 2 1 75: 3 2 2 1 110: 3 3 1 143: 2 2 1
%e 40: 4 2 2 1 76: 3 2 2 1 111: 2 2 1 144: 6 2 2 1
%e 42: 3 3 1 77: 2 2 1 112: 5 2 2 1 145: 2 2 1
%e 44: 3 2 2 1 78: 3 3 1 114: 3 3 1 146: 2 2 1
%e 45: 3 2 2 1 80: 5 2 2 1 115: 2 2 1 147: 3 2 2 1
%e 46: 2 2 1 82: 2 2 1 116: 3 2 2 1 148: 3 2 2 1
%e 48: 5 2 2 1 84: 4 3 2 2 1 117: 3 2 2 1 150: 4 3 2 2 1
%e 50: 3 2 2 1 85: 2 2 1 118: 2 2 1 152: 4 2 2 1
%t omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
%t Select[Range[100],!UnsameQ@@omseq[#]&]
%Y Positions of nonsquarefree numbers in A325248.
%Y Cf. A056239, A112798, A118914, A181819, A323023, A325247, A325249, A325250, A325251, A325277, A325285.
%Y Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (frequency depth), A325248 (Heinz number), A325249 (sum).
%K nonn
%O 1,1
%A _Gus Wiseman_, Apr 24 2019
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