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

1,1

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.

Links

Table of n, a(n) for n=1..49.

Example

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}

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&]

Crossrefs

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

Sequence in context: A363814 A224218 A076485 * A272860 A071354 A006622

Adjacent sequences: A325384 A325385 A325386 * A325388 A325389 A325390

Keyword

nonn

Author

Gus Wiseman, May 02 2019

Status

approved