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A325374
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Numbers with adjusted frequency depth 3 whose prime indices cover an initial interval of positive integers.
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4
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6, 30, 36, 210, 216, 900, 1296, 2310, 7776, 27000, 30030, 44100, 46656, 279936, 510510, 810000, 1679616, 5336100, 9261000, 9699690, 10077696, 24300000, 60466176, 223092870, 362797056, 729000000, 901800900, 1944810000, 2176782336, 6469693230, 12326391000
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OFFSET
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1,1
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COMMENTS
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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.
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.
The terms are the primorial numbers (A002110) above 2 and all their powers. - Amiram Eldar, May 08 2019
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LINKS
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EXAMPLE
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The sequence of terms together with their prime indices begins:
6: {1,2}
30: {1,2,3}
36: {1,1,2,2}
210: {1,2,3,4}
216: {1,1,1,2,2,2}
900: {1,1,2,2,3,3}
1296: {1,1,1,1,2,2,2,2}
2310: {1,2,3,4,5}
7776: {1,1,1,1,1,2,2,2,2,2}
27000: {1,1,1,2,2,2,3,3,3}
30030: {1,2,3,4,5,6}
44100: {1,1,2,2,3,3,4,4}
46656: {1,1,1,1,1,1,2,2,2,2,2,2}
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MATHEMATICA
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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[#]==3&]
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CROSSREFS
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Cf. A002110, A055932, A056239, A112798, A130091, A181819, A320348, A323014, A325280, A325326, A325336, A325387.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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