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

12, 18, 24, 48, 54, 72, 96, 108, 144, 162, 192, 288, 324, 360, 384, 432, 486, 540, 576, 600, 648, 720, 768, 864, 972, 1152, 1200, 1260, 1350, 1440, 1458, 1500, 1536, 1620, 1728, 1944, 2100, 2160, 2250, 2304, 2400, 2592, 2880, 2916, 2940, 3072, 3150, 3240, 3456

Offset: 1

Author

Gus Wiseman, May 02 2019

Keywords

• nonn

Comments

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.

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.

Examples

			The sequence of terms together with their prime indices begins:
    12: {1,1,2}
    18: {1,2,2}
    24: {1,1,1,2}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    72: {1,1,1,2,2}
    96: {1,1,1,1,1,2}
   108: {1,1,2,2,2}
   144: {1,1,1,1,2,2}
   162: {1,2,2,2,2}
   192: {1,1,1,1,1,1,2}
   288: {1,1,1,1,1,2,2}
   324: {1,1,2,2,2,2}
   360: {1,1,1,2,2,3}
   384: {1,1,1,1,1,1,1,2}
   432: {1,1,1,1,2,2,2}
   486: {1,2,2,2,2,2}
   540: {1,1,2,2,2,3}
   576: {1,1,1,1,1,1,2,2}
   600: {1,1,1,2,3,3}
		

Crossrefs

Cf. A055932, A056239, A112798, A181819,, A323014, A325280, A325326, A325335, A325336, A325374.

Programs

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[#]==4&]