

A325281


Numbers of the form a*b, a*a*b, or a*a*b*c where a, b, and c are distinct primes. Numbers with sorted prime signature (1,1), (1,2), or (1,1,2).


5



6, 10, 12, 14, 15, 18, 20, 21, 22, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 50, 51, 52, 55, 57, 58, 60, 62, 63, 65, 68, 69, 74, 75, 76, 77, 82, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 98, 99, 106, 111, 115, 116, 117, 118, 119, 122, 123, 124, 126, 129, 132
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OFFSET

1,1


COMMENTS

Also numbers whose adjusted frequency depth is one plus their number of prime factors counted with multiplicity. The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is one plus the number of times one must apply A181819 to reach a prime number, where A181819(k = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of k. 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 plus 1. The enumeration of these partitions by sum is given by A127002.


LINKS



EXAMPLE

The sequence of terms together with their prime indices and their omegasequences (see A323023) begins:
6: {1,2} (2,2,1)
10: {1,3} (2,2,1)
12: {1,1,2} (3,2,2,1)
14: {1,4} (2,2,1)
15: {2,3} (2,2,1)
18: {1,2,2} (3,2,2,1)
20: {1,1,3} (3,2,2,1)
21: {2,4} (2,2,1)
22: {1,5} (2,2,1)
26: {1,6} (2,2,1)
28: {1,1,4} (3,2,2,1)
33: {2,5} (2,2,1)
34: {1,7} (2,2,1)
35: {3,4} (2,2,1)
38: {1,8} (2,2,1)
39: {2,6} (2,2,1)
44: {1,1,5} (3,2,2,1)
45: {2,2,3} (3,2,2,1)
46: {1,9} (2,2,1)
50: {1,3,3} (3,2,2,1)
51: {2,7} (2,2,1)
52: {1,1,6} (3,2,2,1)
55: {3,5} (2,2,1)
57: {2,8} (2,2,1)
58: {1,10} (2,2,1)
60: {1,1,2,3} (4,3,2,2,1)


MATHEMATICA

fdadj[n_Integer]:=If[n==1, 0, Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#]&, n, !PrimeQ[#]&]]];
Select[Range[100], fdadj[#]==PrimeOmega[#]+1&]


CROSSREFS

Cf. A056239, A112798, A118914, A127002, A181819, A323023, A325246, A325259, A325266, A325270, A325277, A325282.


KEYWORD

nonn


AUTHOR



STATUS

approved



