A323055 Numbers with exactly two distinct exponents in their prime factorization, or two distinct parts in their prime signature.

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200

Offset: 1

Author

Gus Wiseman, Jan 03 2019

Keywords

• nonn

Comments

The first term is A006939(2) = 12.

First differs from A059404 in lacking 360, whose prime signature has three distinct parts.

Positions of 2's in A071625.

Numbers k such that A001221(A181819(k)) = 2.

The asymptotic density of this sequence is (6/Pi^2) * Sum_{n>=2, n squarefree} 1/((n-1)*psi(n)) = 0.3611398..., where psi is the Dedekind psi function (A001615) (Sanna, 2020). - Amiram Eldar, Oct 18 2020

Examples

			3000 = 2^3 * 3^1 * 5^3 has two distinct exponents {1, 3}, 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

One distinct exponent: A062770 or A072774.

Two distinct exponents: this sequence.

Three distinct exponents: A323024.

Four distinct exponents: A323025.

Five distinct exponents: A323056.

Cf. A001221, A001222, A001615, A006939, A007774, A059404, A071625, A118914, A181819, A323014, A323022.

Programs

Maple

isA323055 := proc(n)
    local eset;
    eset := {};
    for pf in ifactors(n)[2] do
        eset := eset union {pf[2]} ;
    end do:
    simplify(nops(eset) = 2 ) ;
end proc:
for n from 12 to 1000 do
    if isA323055(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 09 2019

Mathematica

Select[Range[100],Length[Union[Last/@FactorInteger[#]]]==2&]