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

1,1

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, 1<d<n} 1/(d-1) (Sanna, 2020). - Amiram Eldar, Oct 18 2020

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.

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A137487 A069478 A060665 * A072414 A163569 A063067

Adjacent sequences: A323021 A323022 A323023 * A323025 A323026 A323027

Keyword

nonn

Author

Gus Wiseman, Jan 02 2019

Status

approved