A325266 - OEIS

%I #6 Apr 17 2019 19:08:58

%S 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,

%T 54,56,59,61,66,67,70,71,73,78,79,83,88,89,97,101,102,103,104,105,107,

%U 109,110,113,114,120,121,127,130,131,135,136,137,138,139,149

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

%C 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.

%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. The enumeration of these partitions by sum is given by A325246.

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

%e    2:       {1} (1)

%e    3:       {2} (1)

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

%e    5:       {3} (1)

%e    7:       {4} (1)

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

%e   11:       {5} (1)

%e   13:       {6} (1)

%e   17:       {7} (1)

%e   19:       {8} (1)

%e   23:       {9} (1)

%e   24: {1,1,1,2} (4,2,2,1)

%e   25:     {3,3} (2,1)

%e   29:      {10} (1)

%e   30:   {1,2,3} (3,3,1)

%e   31:      {11} (1)

%e   37:      {12} (1)

%e   40: {1,1,1,3} (4,2,2,1)

%e   41:      {13} (1)

%e   42:   {1,2,4} (3,3,1)

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

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

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

%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).

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 17 2019