A325374 - OEIS

%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