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proposed

allocated for Gus WisemanNumbers 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

1,1

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.

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}

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&]

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

allocated

nonn

Gus Wiseman, May 02 2019

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editing

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allocated for Gus Wiseman

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