A325266 Numbers whose adjusted frequency depth equals their number of prime factors counted with multiplicity.

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

Offset: 1

Author

Gus Wiseman, Apr 17 2019

Keywords

• nonn

Comments

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.

Examples

			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)
		

Crossrefs

Cf. A056239, A112798, A118914, A181819, A225485, A323023, A325246, A325258, A325277, A325278, A325281, A325283.

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).

Programs

Mathematica

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