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

Author

Gus Wiseman, Jan 02 2019

Keywords

• nonn

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, 1Amiram Eldar, Oct 18 2020

Examples

			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.
		

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.

Crossrefs

Cf. A001221, A001222, A001615, A006939, A033993, A059404, A062770, A071625, A118914, A181819, A323014, A323022, A323024.

Programs

Mathematica

tom[n_]:=Length[Union[Last/@If[n==1,{},FactorInteger[n]]]];
Select[Range[100000],tom[#]==4&]

PARI

is(n) = #Set(factor(n)[, 2]) == 4 \\ David A. Corneth, Jan 09 2019