OFFSET
1,1
COMMENTS
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.
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 24 2019
STATUS
approved