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A323025 Numbers with exactly four distinct exponents in their prime factorization, or four distinct parts in their prime signature. 8
75600, 105840, 113400, 118800, 126000, 140400, 151200, 158760, 178200, 183600, 198000, 205200, 210600, 211680, 232848, 234000, 237600, 246960, 248400, 252000, 261360, 275184, 275400, 280800, 283500, 294000, 302400, 306000, 307800, 313200, 315000, 334800 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Positions of 4's in A071625.

Numbers k such that A001221(A181819(k)) = 4.

Is a(n) ~ c * n for some c? - David A. Corneth, Jan 09 2019

The asymptotic density of this sequence is (6/Pi^2) * Sum_{n>=2, n squarefree} r(n)/((n-1)*psi(n)) = 0.00035750... (corresponding to c = 2797.1... in the question above, whose answer is affirmative), where psi is the Dedekind psi function (A001615), and r(n) = Sum_{d_1|n, 1<d_1<n} (1/(d_1-1)) * Sum_{d_2|d_1, 1<d_2<d_1} 1/(d_2-1) (Sanna, 2020). - Amiram Eldar, Oct 18 2020

LINKS

David A. Corneth, Table of n, a(n) for n = 1..10000

Carlo Sanna, On the number of distinct exponents in the prime factorization of an integer, Proceedings - Mathematical Sciences, Indian Academy of Sciences, Vol. 130, No. 1 (2020), Article 27, alternative link.

EXAMPLE

126000 = 2^4 * 3^2 * 5^3 * 7^1 has four distinct exponents {1, 2, 3, 4}, so belongs to the sequence.

831600 = 2^4 * 3^3 * 5^2 * 7^1 * 11^1 has four distinct exponents {1, 2, 3, 4}, so belongs to the sequence.

MATHEMATICA

tom[n_]:=Length[Union[Last/@If[n==1, {}, FactorInteger[n]]]];

Select[Range[100000], tom[#]==4&]

PROG

(PARI) is(n) = #Set(factor(n)[, 2]) == 4 \\ David A. Corneth, Jan 09 2019

CROSSREFS

Cf. A001221, A001222, A001615, A006939, A033993, A059404, A062770, A071625, A118914, A181819, A323014, A323022, A323024.

Sequence in context: A236817 A140940 A204865 * A323804 A135209 A243959

Adjacent sequences:  A323022 A323023 A323024 * A323026 A323027 A323028

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 02 2019

STATUS

approved

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Last modified July 7 12:08 EDT 2022. Contains 355148 sequences. (Running on oeis4.)