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%I #5 May 02 2019 16:05:46
%S 12,18,24,48,54,72,96,108,144,162,192,288,324,360,384,432,486,540,576,
%T 600,648,720,768,864,972,1152,1200,1260,1350,1440,1458,1500,1536,1620,
%U 1728,1944,2100,2160,2250,2304,2400,2592,2880,2916,2940,3072,3150,3240,3456
%N Numbers with adjusted frequency depth 4 whose prime indices cover an initial interval of positive integers.
%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(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 4 whose parts cover an initial interval of positive integers. The enumeration of these partitions by sum is given by A325335.
%e The sequence of terms together with their prime indices begins:
%e 12: {1,1,2}
%e 18: {1,2,2}
%e 24: {1,1,1,2}
%e 48: {1,1,1,1,2}
%e 54: {1,2,2,2}
%e 72: {1,1,1,2,2}
%e 96: {1,1,1,1,1,2}
%e 108: {1,1,2,2,2}
%e 144: {1,1,1,1,2,2}
%e 162: {1,2,2,2,2}
%e 192: {1,1,1,1,1,1,2}
%e 288: {1,1,1,1,1,2,2}
%e 324: {1,1,2,2,2,2}
%e 360: {1,1,1,2,2,3}
%e 384: {1,1,1,1,1,1,1,2}
%e 432: {1,1,1,1,2,2,2}
%e 486: {1,2,2,2,2,2}
%e 540: {1,1,2,2,2,3}
%e 576: {1,1,1,1,1,1,2,2}
%e 600: {1,1,1,2,3,3}
%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[#]==4&]
%Y Cf. A055932, A056239, A112798, A181819,, A323014, A325280, A325326, A325335, A325336, A325374.
%K nonn
%O 1,1
%A _Gus Wiseman_, May 02 2019
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