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).

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

Offset: 1

Author

Gus Wiseman, Apr 18 2019

Keywords

• nonn

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.

Examples

			The sequence of terms together with their prime indices and their omega-sequences (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)
		

Crossrefs

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

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).

Programs

Mathematica

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