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

1,1

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

Table of n, a(n) for n=1..61.

Example

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)

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.

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).

Sequence in context: A237051 A340749 A296205 * A100658 A182301 A069059

Adjacent sequences: A325278 A325279 A325280 * A325282 A325283 A325284

Keyword

nonn

Author

Gus Wiseman, Apr 18 2019

Status

approved