%I #12 May 08 2019 15:53:13
%S 6,30,36,210,216,900,1296,2310,7776,27000,30030,44100,46656,279936,
%T 510510,810000,1679616,5336100,9261000,9699690,10077696,24300000,
%U 60466176,223092870,362797056,729000000,901800900,1944810000,2176782336,6469693230,12326391000
%N Numbers with adjusted frequency depth 3 whose prime indices cover an initial interval of positive integers.
%C 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.
%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 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.
%C The terms are the primorial numbers (A002110) above 2 and all their powers. - _Amiram Eldar_, May 08 2019
%H Amiram Eldar, <a href="/A325374/b325374.txt">Table of n, a(n) for n = 1..5292</a>
%e The sequence of terms together with their prime indices begins:
%e 6: {1,2}
%e 30: {1,2,3}
%e 36: {1,1,2,2}
%e 210: {1,2,3,4}
%e 216: {1,1,1,2,2,2}
%e 900: {1,1,2,2,3,3}
%e 1296: {1,1,1,1,2,2,2,2}
%e 2310: {1,2,3,4,5}
%e 7776: {1,1,1,1,1,2,2,2,2,2}
%e 27000: {1,1,1,2,2,2,3,3,3}
%e 30030: {1,2,3,4,5,6}
%e 44100: {1,1,2,2,3,3,4,4}
%e 46656: {1,1,1,1,1,1,2,2,2,2,2,2}
%t normQ[n_Integer]:=Or[n==1,PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
%t fdadj[n_Integer]:=If[n==1,0,Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#1]&,n,!PrimeQ[#1]&]]];
%t Select[Range[10000],normQ[#]&&fdadj[#]==3&]
%Y Cf. A002110, A055932, A056239, A112798, A130091, A181819, A320348, A323014, A325280, A325326, A325336, A325387.
%K nonn
%O 1,1
%A _Gus Wiseman_, May 02 2019