A323024 Numbers with exactly three distinct exponents in their prime factorization, or three distinct parts in their prime signature.

360, 504, 540, 600, 720, 756, 792, 936, 1008, 1176, 1188, 1200, 1224, 1350, 1368, 1400, 1404, 1440, 1500, 1584, 1620, 1656, 1836, 1872, 1960, 2016, 2052, 2088, 2160, 2200, 2232, 2250, 2268, 2352, 2400, 2448, 2484, 2520, 2600, 2646, 2664, 2736, 2800, 2880, 2904

Offset: 1

Author

Gus Wiseman, Jan 02 2019

Keywords

• nonn

Comments

Positions of 3's in A071625.

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

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, 1Amiram Eldar, Oct 18 2020

Examples

			1500 = 2^2 * 3^1 * 5^3 has three distinct exponents {1, 2, 3}, so belongs to the sequence.
52500 = 2^2 * 3^1 * 5^4 * 7^1 has three distinct exponents {1, 2, 4}, so belongs to the sequence.
		

Links

• Amiram Eldar, 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.

Crossrefs

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

Programs

Mathematica

tom[n_]:=Length[Union[Last/@If[n==1,{},FactorInteger[n]]]];
Select[Range[1000],tom[#]==3&]

PARI

is(n) = #Set(factor(n)[, 2]) == 3 \\ David A. Corneth, Jan 02 2019