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A079275 Number of divisors of n that are semiprimes with distinct factors.

%I A079275 #27 Jul 06 2025 11:11:04
%S A079275 0,0,0,0,0,1,0,0,0,1,0,1,0,1,1,0,0,1,0,1,1,1,0,1,0,1,0,1,0,3,0,0,1,1,
%T A079275 1,1,0,1,1,1,0,3,0,1,1,1,0,1,0,1,1,1,0,1,1,1,1,1,0,3,0,1,1,0,1,3,0,1,
%U A079275 1,3,0,1,0,1,1,1,1,3,0,1,0,1,0,3,1,1,1,1,0,3,1,1,1,1,1,1,0,1,1,1,0,3,0,1,3
%N A079275 Number of divisors of n that are semiprimes with distinct factors.
%C A079275 Number of pairs of prime factors of n, (p,q), such that p < q. For example, the prime factors of 30 are 2, 3 and 5, so we have the ordered pairs (2,3), (2,5) and (3,5). - _Wesley Ivan Hurt_, Sep 14 2020
%C A079275 Inverse Möbius transform of A280710(n). - _Wesley Ivan Hurt_, Jul 06 2025
%H A079275 G. C. Greubel, <a href="/A079275/b079275.txt">Table of n, a(n) for n = 1..1000</a>
%F A079275 a(A000961(n)) = 0; a(A007774(n)) = 1; a(A033992(n)) = 3; a(A033993(n)) = 6.
%F A079275 a(n) = omega(n)*(omega(n)-1)/2, where omega(n) is the number of distinct prime factors of n.
%F A079275 a(n) = Sum_{p|n, q|n, p,q prime, p<q} 1. - _Wesley Ivan Hurt_, Sep 14 2020
%F A079275 a(n) = Sum_{d|n} A280710(d). - _Wesley Ivan Hurt_, Jul 06 2025
%p A079275 A079275 := proc(n)
%p A079275     local a,d ;
%p A079275     a := 0 ;
%p A079275     for d in numtheory[divisors](n) do
%p A079275         if A001221(d) = 2 and A001222(d) = 2 then
%p A079275             a := a+1 ;
%p A079275         end if;
%p A079275     end do:
%p A079275     a ;
%p A079275 end proc:
%p A079275 seq(A079275(n),n=1..40) ; # _R. J. Mathar_, Jan 18 2021
%t A079275 f[n_]:=Module[{c=PrimeNu[n]},(c(c-1))/2]; Array[f,110] (* _Harvey P. Dale_, Oct 05 2011 *)
%o A079275 (PARI) a(n) = sumdiv(n, d, (bigomega(d)==2) && (omega(d)==2)); \\ _Michel Marcus_, Sep 15 2020
%o A079275 (PARI) a(n) = binomial(omega(n),2) \\ _David A. Corneth_, Sep 15 2020
%Y A079275 Cf. A000005, A000961, A001221, A001358, A006881, A007774, A033992, A033993, A280710.
%K A079275 nonn,easy
%O A079275 1,30
%A A079275 _Reinhard Zumkeller_, Feb 07 2003