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

1,30

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

Links

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

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, p<q} 1. - Wesley Ivan Hurt, Sep 14 2020

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A100655 A377381 A262262 * A236314 A236322 A319419

Adjacent sequences: A079272 A079273 A079274 * A079276 A079277 A079278

Keyword

nonn,easy

Author

Reinhard Zumkeller, Feb 07 2003

Status

approved