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A325387 Numbers with adjusted frequency depth 4 whose prime indices cover an initial interval of positive integers. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A245379 A224218 A076485 * A272860 A071354 A006622

Adjacent sequences:  A325384 A325385 A325386 * A325388 A325389 A325390

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 02 2019

STATUS

approved

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Last modified January 25 16:42 EST 2020. Contains 331245 sequences. (Running on oeis4.)