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A323055
Numbers with exactly two distinct exponents in their prime factorization, or two distinct parts in their prime signature.
5
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
OFFSET
1,1
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
LINKS
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
3000 = 2^3 * 3^1 * 5^3 has two distinct exponents {1, 3}, so belongs to the sequence.
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&]
CROSSREFS
One distinct exponent: A062770 or A072774.
Two distinct exponents: A323055.
Three distinct exponents: A323024.
Four distinct exponents: A323025.
Five distinct exponents: A323056.
Sequence in context: A369516 A317711 A359890 * A059404 A376250 A303946
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 03 2019
STATUS
approved