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A325266
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Numbers whose adjusted frequency depth equals their number of prime factors counted with multiplicity.
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2
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1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 24, 25, 29, 30, 31, 37, 40, 41, 42, 43, 47, 49, 53, 54, 56, 59, 61, 66, 67, 70, 71, 73, 78, 79, 83, 88, 89, 97, 101, 102, 103, 104, 105, 107, 109, 110, 113, 114, 120, 121, 127, 130, 131, 135, 136, 137, 138, 139, 149
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OFFSET
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1,2
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COMMENTS
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The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is 1 plus the number of times one must apply A181819 to reach a prime number, where A181819(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, 180 has adjusted frequency depth 5 because we have: 180 -> 18 -> 6 -> 4 -> 3.
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 adjusted frequency depth is equal to their length. The enumeration of these partitions by sum is given by A325246.
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LINKS
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EXAMPLE
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The sequence of terms together with their prime indices and their omega-sequences (see A323023) begins:
2: {1} (1)
3: {2} (1)
4: {1,1} (2,1)
5: {3} (1)
7: {4} (1)
9: {2,2} (2,1)
11: {5} (1)
13: {6} (1)
17: {7} (1)
19: {8} (1)
23: {9} (1)
24: {1,1,1,2} (4,2,2,1)
25: {3,3} (2,1)
29: {10} (1)
30: {1,2,3} (3,3,1)
31: {11} (1)
37: {12} (1)
40: {1,1,1,3} (4,2,2,1)
41: {13} (1)
42: {1,2,4} (3,3,1)
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MATHEMATICA
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fdadj[n_Integer]:=If[n==1, 0, Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#]&, n, !PrimeQ[#]&]]];
Select[Range[100], fdadj[#]==PrimeOmega[#]&]
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CROSSREFS
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Cf. A056239, A112798, A118914, A181819, A225485, A323023, A325246, A325258, A325277, A325278, A325281, A325283.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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