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 A325411 Numbers whose omega-sequence has repeated parts. 1
 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS First differs from A323304 in lacking 216. First differs from A106543 in having 144. 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. 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. LINKS EXAMPLE The sequence of terms together with their omega-sequences begins: 6: 2 2 1 51: 2 2 1 86: 2 2 1 119: 2 2 1 10: 2 2 1 52: 3 2 2 1 87: 2 2 1 120: 5 3 2 2 1 12: 3 2 2 1 54: 4 2 2 1 88: 4 2 2 1 122: 2 2 1 14: 2 2 1 55: 2 2 1 90: 4 3 2 2 1 123: 2 2 1 15: 2 2 1 56: 4 2 2 1 91: 2 2 1 124: 3 2 2 1 18: 3 2 2 1 57: 2 2 1 92: 3 2 2 1 126: 4 3 2 2 1 20: 3 2 2 1 58: 2 2 1 93: 2 2 1 129: 2 2 1 21: 2 2 1 60: 4 3 2 2 1 94: 2 2 1 130: 3 3 1 22: 2 2 1 62: 2 2 1 95: 2 2 1 132: 4 3 2 2 1 24: 4 2 2 1 63: 3 2 2 1 96: 6 2 2 1 133: 2 2 1 26: 2 2 1 65: 2 2 1 98: 3 2 2 1 134: 2 2 1 28: 3 2 2 1 66: 3 3 1 99: 3 2 2 1 135: 4 2 2 1 30: 3 3 1 68: 3 2 2 1 102: 3 3 1 136: 4 2 2 1 33: 2 2 1 69: 2 2 1 104: 4 2 2 1 138: 3 3 1 34: 2 2 1 70: 3 3 1 105: 3 3 1 140: 4 3 2 2 1 35: 2 2 1 72: 5 2 2 1 106: 2 2 1 141: 2 2 1 38: 2 2 1 74: 2 2 1 108: 5 2 2 1 142: 2 2 1 39: 2 2 1 75: 3 2 2 1 110: 3 3 1 143: 2 2 1 40: 4 2 2 1 76: 3 2 2 1 111: 2 2 1 144: 6 2 2 1 42: 3 3 1 77: 2 2 1 112: 5 2 2 1 145: 2 2 1 44: 3 2 2 1 78: 3 3 1 114: 3 3 1 146: 2 2 1 45: 3 2 2 1 80: 5 2 2 1 115: 2 2 1 147: 3 2 2 1 46: 2 2 1 82: 2 2 1 116: 3 2 2 1 148: 3 2 2 1 48: 5 2 2 1 84: 4 3 2 2 1 117: 3 2 2 1 150: 4 3 2 2 1 50: 3 2 2 1 85: 2 2 1 118: 2 2 1 152: 4 2 2 1 MATHEMATICA omseq[n_Integer]:=If[n<=1, {}, Total/@NestWhileList[Sort[Length/@Split[#]]&, Sort[Last/@FactorInteger[n]], Total[#]>1&]]; Select[Range[100], !UnsameQ@@omseq[#]&] CROSSREFS Positions of nonsquarefree numbers in A325248. Cf. A056239, A112798, A118914, A181819, A323023, A325247, A325249, A325250, A325251, A325277, A325285. 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). Sequence in context: A361102 A335080 A323304 * A106543 A324455 A327476 Adjacent sequences: A325408 A325409 A325410 * A325412 A325413 A325414 KEYWORD nonn AUTHOR Gus Wiseman, Apr 24 2019 STATUS approved

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Last modified March 24 17:57 EDT 2023. Contains 361506 sequences. (Running on oeis4.)