cp's OEIS Frontend

A195086 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 2.

%I A195086 #40 Apr 10 2025 06:50:58
%S A195086 8,24,27,36,40,54,56,88,100,104,120,125,135,136,152,168,180,184,189,
%T A195086 196,225,232,248,250,252,264,270,280,296,297,300,312,328,343,344,351,
%U A195086 375,376,378,396,408,424,440,441,450,456,459,468,472,484,488
%N A195086 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 2.
%C A195086 From _Amiram Eldar_, Nov 07 2020: (Start)
%C A195086 Numbers whose powerful part (A057521) is either a cube of a prime (A030078) or a square of a squarefree semiprime (A085986).
%C A195086 The asymptotic density of this sequence is (6/Pi^2) * (Sum_{p prime} 1/(p^2*(p+1)) + Sum_{p<q primes} 1/(p*(p+1)*q*(q+1))) = (1/zeta(2)) * (2*P(3) + Sum_{k>=4} (-1)^(k+1)*(k-1)*P(k) + (Sum_{k>=2} (-1)^k*P(k))^2)/2 = 0.0963023158..., where P is the prime zeta function. (End)
%H A195086 Reinhard Zumkeller, <a href="/A195086/b195086.txt">Table of n, a(n) for n = 1..10000</a>
%H A195086 Alfred Rényi, <a href="http://elib.mi.sanu.ac.rs/files/journals/publ/14/13.pdf">On the density of certain sequences of integers</a>, Publications de l'Institut Mathématique, Vol. 8 (1955), pp. 157-162.
%H A195086 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A195086 A001222(a(n)) - A001221(a(n)) = 2.
%F A195086 A046660(a(n)) = 2. - _Reinhard Zumkeller_, Nov 29 2015
%t A195086 Select[Range[500],PrimeOmega[#]-PrimeNu[#]==2&]
%o A195086 (PARI) is(n)=bigomega(n)-omega(n)==2 \\ _Charles R Greathouse IV_, Sep 14 2015
%o A195086 (PARI) is(n)=my(f=factor(n)[,2]); vecsum(f)==#f+2 \\ _Charles R Greathouse IV_, Aug 01 2016
%o A195086 (Haskell)
%o A195086 a195086 n = a195086_list !! (n-1)
%o A195086 a195086_list = filter ((== 2) . a046660) [1..]
%o A195086 -- _Reinhard Zumkeller_, Nov 29 2015
%Y A195086 Subsequence of A048108.
%Y A195086 Subsequences: A030078 and A085986.
%Y A195086 Cf. A001221, A001222, A025487, A057521, A060687, A195069, A195087, A195088, A195089, A195090, A195091, A195092, A195093, A046660, A257851, A261256, A264959.
%K A195086 nonn
%O A195086 1,1
%A A195086 _Harvey P. Dale_, Sep 08 2011