OFFSET
1,1
COMMENTS
The adjusted frequency depth (A323014) 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(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 with adjusted frequency depth 3 whose parts cover an initial interval of positive integers. The enumeration of these partitions by sum is given by A325334.
The terms are the primorial numbers (A002110) above 2 and all their powers. - Amiram Eldar, May 08 2019
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..5292
EXAMPLE
The sequence of terms together with their prime indices begins:
6: {1,2}
30: {1,2,3}
36: {1,1,2,2}
210: {1,2,3,4}
216: {1,1,1,2,2,2}
900: {1,1,2,2,3,3}
1296: {1,1,1,1,2,2,2,2}
2310: {1,2,3,4,5}
7776: {1,1,1,1,1,2,2,2,2,2}
27000: {1,1,1,2,2,2,3,3,3}
30030: {1,2,3,4,5,6}
44100: {1,1,2,2,3,3,4,4}
46656: {1,1,1,1,1,1,2,2,2,2,2,2}
MATHEMATICA
normQ[n_Integer]:=Or[n==1, PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
fdadj[n_Integer]:=If[n==1, 0, Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#1]&, n, !PrimeQ[#1]&]]];
Select[Range[10000], normQ[#]&&fdadj[#]==3&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 02 2019
STATUS
approved