cp's OEIS Frontend

A325387 Numbers with adjusted frequency depth 4 whose prime indices cover an initial interval of positive integers.

%I A325387 #5 May 02 2019 16:05:46
%S A325387 12,18,24,48,54,72,96,108,144,162,192,288,324,360,384,432,486,540,576,
%T A325387 600,648,720,768,864,972,1152,1200,1260,1350,1440,1458,1500,1536,1620,
%U A325387 1728,1944,2100,2160,2250,2304,2400,2592,2880,2916,2940,3072,3150,3240,3456
%N A325387 Numbers with adjusted frequency depth 4 whose prime indices cover an initial interval of positive integers.
%C A325387 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 A325387 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 A325387 The sequence of terms together with their prime indices begins:
%e A325387     12: {1,1,2}
%e A325387     18: {1,2,2}
%e A325387     24: {1,1,1,2}
%e A325387     48: {1,1,1,1,2}
%e A325387     54: {1,2,2,2}
%e A325387     72: {1,1,1,2,2}
%e A325387     96: {1,1,1,1,1,2}
%e A325387    108: {1,1,2,2,2}
%e A325387    144: {1,1,1,1,2,2}
%e A325387    162: {1,2,2,2,2}
%e A325387    192: {1,1,1,1,1,1,2}
%e A325387    288: {1,1,1,1,1,2,2}
%e A325387    324: {1,1,2,2,2,2}
%e A325387    360: {1,1,1,2,2,3}
%e A325387    384: {1,1,1,1,1,1,1,2}
%e A325387    432: {1,1,1,1,2,2,2}
%e A325387    486: {1,2,2,2,2,2}
%e A325387    540: {1,1,2,2,2,3}
%e A325387    576: {1,1,1,1,1,1,2,2}
%e A325387    600: {1,1,1,2,3,3}
%t A325387 normQ[n_Integer]:=Or[n==1,PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
%t A325387 fdadj[n_Integer]:=If[n==1,0,Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#1]&,n,!PrimeQ[#1]&]]];
%t A325387 Select[Range[10000],normQ[#]&&fdadj[#]==4&]
%Y A325387 Cf. A055932, A056239, A112798, A181819,, A323014, A325280, A325326, A325335, A325336, A325374.
%K A325387 nonn
%O A325387 1,1
%A A325387 _Gus Wiseman_, May 02 2019