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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

%I

%S 6,10,12,14,15,18,20,21,22,26,28,33,34,35,38,39,44,45,46,50,51,52,55,

%T 57,58,60,62,63,65,68,69,74,75,76,77,82,84,85,86,87,90,91,92,93,94,95,

%U 98,99,106,111,115,116,117,118,119,122,123,124,126,129,132

%N 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).

%C 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.

%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 adjusted frequency depth is equal to their length plus 1. The enumeration of these partitions by sum is given by A127002.

%e The sequence of terms together with their prime indices and their omega-sequences (see A323023) begins:

%e 6: {1,2} (2,2,1)

%e 10: {1,3} (2,2,1)

%e 12: {1,1,2} (3,2,2,1)

%e 14: {1,4} (2,2,1)

%e 15: {2,3} (2,2,1)

%e 18: {1,2,2} (3,2,2,1)

%e 20: {1,1,3} (3,2,2,1)

%e 21: {2,4} (2,2,1)

%e 22: {1,5} (2,2,1)

%e 26: {1,6} (2,2,1)

%e 28: {1,1,4} (3,2,2,1)

%e 33: {2,5} (2,2,1)

%e 34: {1,7} (2,2,1)

%e 35: {3,4} (2,2,1)

%e 38: {1,8} (2,2,1)

%e 39: {2,6} (2,2,1)

%e 44: {1,1,5} (3,2,2,1)

%e 45: {2,2,3} (3,2,2,1)

%e 46: {1,9} (2,2,1)

%e 50: {1,3,3} (3,2,2,1)

%e 51: {2,7} (2,2,1)

%e 52: {1,1,6} (3,2,2,1)

%e 55: {3,5} (2,2,1)

%e 57: {2,8} (2,2,1)

%e 58: {1,10} (2,2,1)

%e 60: {1,1,2,3} (4,3,2,2,1)

%t fdadj[n_Integer]:=If[n==1,0,Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#]&,n,!PrimeQ[#]&]]];

%t Select[Range[100],fdadj[#]==PrimeOmega[#]+1&]

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

%Y Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

%K nonn

%O 1,1

%A _Gus Wiseman_, Apr 18 2019

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Last modified November 12 22:16 EST 2019. Contains 329079 sequences. (Running on oeis4.)