cp's OEIS Frontend

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

%I A323025 #22 Oct 18 2020 03:12:39
%S A323025 75600,105840,113400,118800,126000,140400,151200,158760,178200,183600,
%T A323025 198000,205200,210600,211680,232848,234000,237600,246960,248400,
%U A323025 252000,261360,275184,275400,280800,283500,294000,302400,306000,307800,313200,315000,334800
%N A323025 Numbers with exactly four distinct exponents in their prime factorization, or four distinct parts in their prime signature.
%C A323025 Positions of 4's in A071625.
%C A323025 Numbers k such that A001221(A181819(k)) = 4.
%C A323025 Is a(n) ~ c * n for some c? - _David A. Corneth_, Jan 09 2019
%C A323025 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
%H A323025 David A. Corneth, <a href="/A323025/b323025.txt">Table of n, a(n) for n = 1..10000</a>
%H A323025 Carlo Sanna, <a href="https://doi.org/10.1007/s12044-020-0556-y">On the number of distinct exponents in the prime factorization of an integer</a>, Proceedings - Mathematical Sciences, Indian Academy of Sciences, Vol. 130, No. 1 (2020), Article 27, <a href="https://www.ias.ac.in/describe/article/pmsc/130/0027">alternative link</a>.
%e A323025 126000 = 2^4 * 3^2 * 5^3 * 7^1 has four distinct exponents {1, 2, 3, 4}, so belongs to the sequence.
%e A323025 831600 = 2^4 * 3^3 * 5^2 * 7^1 * 11^1 has four distinct exponents {1, 2, 3, 4}, so belongs to the sequence.
%t A323025 tom[n_]:=Length[Union[Last/@If[n==1,{},FactorInteger[n]]]];
%t A323025 Select[Range[100000],tom[#]==4&]
%o A323025 (PARI) is(n) = #Set(factor(n)[, 2]) == 4 \\ _David A. Corneth_, Jan 09 2019
%Y A323025 Cf. A001221, A001222, A001615, A006939, A033993, A059404, A062770, A071625, A118914, A181819, A323014, A323022, A323024.
%K A323025 nonn
%O A323025 1,1
%A A323025 _Gus Wiseman_, Jan 02 2019