A323024 - OEIS

%I #16 Oct 18 2020 03:11:49

%S 360,504,540,600,720,756,792,936,1008,1176,1188,1200,1224,1350,1368,

%T 1400,1404,1440,1500,1584,1620,1656,1836,1872,1960,2016,2052,2088,

%U 2160,2200,2232,2250,2268,2352,2400,2448,2484,2520,2600,2646,2664,2736,2800,2880,2904

%N Numbers with exactly three distinct exponents in their prime factorization, or three distinct parts in their prime signature.

%C Positions of 3's in A071625.

%C Numbers k such that A001221(A181819(k)) = 3.

%C The asymptotic density of this sequence is (6/Pi^2) * Sum_{n>=2, n squarefree} r(n)/((n-1)*psi(n)) = 0.030575..., where psi is the Dedekind psi function (A001615), and r(n) = Sum_{d|n, 1<d<n} 1/(d-1) (Sanna, 2020). - _Amiram Eldar_, Oct 18 2020

%H Amiram Eldar, <a href="/A323024/b323024.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 1500 = 2^2 * 3^1 * 5^3 has three distinct exponents {1, 2, 3}, so belongs to the sequence.

%e 52500 = 2^2 * 3^1 * 5^4 * 7^1 has three distinct exponents {1, 2, 4}, so belongs to the sequence.

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

%t Select[Range[1000],tom[#]==3&]

%o (PARI) is(n) = #Set(factor(n)[, 2]) == 3 \\ _David A. Corneth_, Jan 02 2019

%Y Cf. A001221, A001222, A001615, A006939, A033992, A059404, A062770, A071625, A118914, A181819, A323014, A323022, A323025.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jan 02 2019