cp's OEIS Frontend

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).

%I A325281 #8 Apr 19 2019 13:42:01
%S A325281 6,10,12,14,15,18,20,21,22,26,28,33,34,35,38,39,44,45,46,50,51,52,55,
%T A325281 57,58,60,62,63,65,68,69,74,75,76,77,82,84,85,86,87,90,91,92,93,94,95,
%U A325281 98,99,106,111,115,116,117,118,119,122,123,124,126,129,132
%N 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).
%C A325281 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 A325281 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 A325281 The sequence of terms together with their prime indices and their omega-sequences (see A323023) begins:
%e A325281    6:     {1,2} (2,2,1)
%e A325281   10:     {1,3} (2,2,1)
%e A325281   12:   {1,1,2} (3,2,2,1)
%e A325281   14:     {1,4} (2,2,1)
%e A325281   15:     {2,3} (2,2,1)
%e A325281   18:   {1,2,2} (3,2,2,1)
%e A325281   20:   {1,1,3} (3,2,2,1)
%e A325281   21:     {2,4} (2,2,1)
%e A325281   22:     {1,5} (2,2,1)
%e A325281   26:     {1,6} (2,2,1)
%e A325281   28:   {1,1,4} (3,2,2,1)
%e A325281   33:     {2,5} (2,2,1)
%e A325281   34:     {1,7} (2,2,1)
%e A325281   35:     {3,4} (2,2,1)
%e A325281   38:     {1,8} (2,2,1)
%e A325281   39:     {2,6} (2,2,1)
%e A325281   44:   {1,1,5} (3,2,2,1)
%e A325281   45:   {2,2,3} (3,2,2,1)
%e A325281   46:     {1,9} (2,2,1)
%e A325281   50:   {1,3,3} (3,2,2,1)
%e A325281   51:     {2,7} (2,2,1)
%e A325281   52:   {1,1,6} (3,2,2,1)
%e A325281   55:     {3,5} (2,2,1)
%e A325281   57:     {2,8} (2,2,1)
%e A325281   58:    {1,10} (2,2,1)
%e A325281   60: {1,1,2,3} (4,3,2,2,1)
%t A325281 fdadj[n_Integer]:=If[n==1,0,Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#]&,n,!PrimeQ[#]&]]];
%t A325281 Select[Range[100],fdadj[#]==PrimeOmega[#]+1&]
%Y A325281 Cf. A056239, A112798, A118914, A127002, A181819, A323023, A325246, A325259, A325266, A325270, A325277, A325282.
%Y A325281 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 A325281 nonn
%O A325281 1,1
%A A325281 _Gus Wiseman_, Apr 18 2019