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

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, 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, 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

Offset: 1

Author

Reinhard Zumkeller, Feb 07 2003

Keywords

• nonn
• easy

Comments

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

Inverse Möbius transform of A280710(n). - Wesley Ivan Hurt, Jul 06 2025

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000005, A000961, A001221, A001358, A006881, A007774, A033992, A033993, A280710.

Programs

Maple

A079275 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if A001221(d) = 2 and A001222(d) = 2 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A079275(n),n=1..40) ; # R. J. Mathar, Jan 18 2021

Mathematica

f[n_]:=Module[{c=PrimeNu[n]},(c(c-1))/2]; Array[f,110] (* Harvey P. Dale, Oct 05 2011 *)

PARI

a(n) = sumdiv(n, d, (bigomega(d)==2) && (omega(d)==2)); \\ Michel Marcus, Sep 15 2020

PARI

a(n) = binomial(omega(n),2) \\ David A. Corneth, Sep 15 2020

Formula

a(A000961(n)) = 0; a(A007774(n)) = 1; a(A033992(n)) = 3; a(A033993(n)) = 6.

a(n) = omega(n)*(omega(n)-1)/2, where omega(n) is the number of distinct prime factors of n.

a(n) = Sum_{p|n, q|n, p,q prime, pWesley Ivan Hurt, Sep 14 2020

a(n) = Sum_{d|n} A280710(d). - Wesley Ivan Hurt, Jul 06 2025