A230595 Number of ways to write n as n = x*y, where x and y are primes, 1 <= x <= n, 1 <= y <= n.

0, 0, 0, 1, 0, 2, 0, 0, 1, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 2, 2

Offset: 1

Author

Jaroslav Krizek, Oct 27 2013

Keywords

• nonn

Comments

Dirichlet convolution of A010051(n) with itself, where A010051 = characteristic function of primes (A000040).

Dirichlet convolution of functions b(n) and c(n) is function a(n) = Sum_(d|n) b(d) * c(n/d).

a(n) = 0, 1 or 2. a(n) = 0 for numbers n from A100959 (non-semiprimes); a(n) = 1 for n = p^2, p = prime; a(n) = 2 for numbers n from A006881 (product of two distinct primes).

Examples

			For n = 6: a(6) = Sum_(d|6) A010051(d) * A010051(6/d) = 0*0 + 1*1 + 1*1 + 1*0 = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000040, A006881, A010051, A100959, A230594.

Programs

Mathematica

Table[Total@ Map[Times @@ Boole@ {PrimeQ@ #, PrimeQ[n/#]} &, FactorInteger[n][[All, 1]]], {n, 95}] (* Michael De Vlieger, Jul 29 2017 *)

PARI

a(n)=sumdiv(n,d,isprime(d)*isprime(n/d)) \\ Ralf Stephan, Oct 30 2013

PARI

a(n) = my(f=factor(f)); (vecsum(f[, 2])==2) * #f~ \\ David A. Corneth, Jul 28 2017

PARI

first(n) = my(v = vector(n)); forprime(p = 2, sqrtint(n), v[p^2] = 1; forprime(q = p + 1, n \ p, v[p*q] = 2)); v \\ David A. Corneth, Jul 28 2017

Python

from sympy import factorint
def A230595(n): return 0 if sum(f:=factorint(n).values())!=2 else len(f) # Chai Wah Wu, Jul 23 2024

Formula

a(n) = Sum_(d|n) A010051(d) * A010051(n/d).

Dirichlet g.f.: primezeta(s)^2. - Benedict W. J. Irwin, Jul 11 2018