A323025 Numbers with exactly four distinct exponents in their prime factorization, or four distinct parts in their prime signature.

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

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