cp's OEIS Frontend

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

%I A323055 #24 Apr 16 2025 04:33:55
%S A323055 12,18,20,24,28,40,44,45,48,50,52,54,56,60,63,68,72,75,76,80,84,88,90,
%T A323055 92,96,98,99,104,108,112,116,117,120,124,126,132,135,136,140,144,147,
%U A323055 148,150,152,153,156,160,162,164,168,171,172,175,176,180,184,188,189,192,198,200
%N A323055 Numbers with exactly two distinct exponents in their prime factorization, or two distinct parts in their prime signature.
%C A323055 The first term is A006939(2) = 12.
%C A323055 First differs from A059404 in lacking 360, whose prime signature has three distinct parts.
%C A323055 Positions of 2's in A071625.
%C A323055 Numbers k such that A001221(A181819(k)) = 2.
%C A323055 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
%H A323055 Amiram Eldar, <a href="/A323055/b323055.txt">Table of n, a(n) for n = 1..10000</a>
%H A323055 Carlo Sanna, <a href="https://doi.org/10.1007/s12044-020-0556-y">On the number of distinct exponents in the prime factorization of an integer</a>, Proceedings - Mathematical Sciences, Indian Academy of Sciences, Vol. 130, No. 1 (2020), Article 27, <a href="https://www.ias.ac.in/describe/article/pmsc/130/0027">alternative link</a>.
%e A323055 3000 = 2^3 * 3^1 * 5^3 has two distinct exponents {1, 3}, so belongs to the sequence.
%p A323055 isA323055 := proc(n)
%p A323055     local eset;
%p A323055     eset := {};
%p A323055     for pf in ifactors(n)[2] do
%p A323055         eset := eset union {pf[2]} ;
%p A323055     end do:
%p A323055     simplify(nops(eset) = 2 ) ;
%p A323055 end proc:
%p A323055 for n from 12 to 1000 do
%p A323055     if isA323055(n) then
%p A323055         printf("%d,",n) ;
%p A323055     end if;
%p A323055 end do: # _R. J. Mathar_, Jan 09 2019
%t A323055 Select[Range[100],Length[Union[Last/@FactorInteger[#]]]==2&]
%Y A323055 One distinct exponent: A062770 or A072774.
%Y A323055 Two distinct exponents: this sequence.
%Y A323055 Three distinct exponents: A323024.
%Y A323055 Four distinct exponents: A323025.
%Y A323055 Five distinct exponents: A323056.
%Y A323055 Cf. A001221, A001222, A001615, A006939, A007774, A059404, A071625, A118914, A181819, A323014, A323022.
%K A323055 nonn
%O A323055 1,1
%A A323055 _Gus Wiseman_, Jan 03 2019