A230594 Number of ways to write n as n = x*y, where x, y = noncomposite numbers (A008578), 1 <= x <= n, 1 <= y <= n.

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

Offset: 1

Author

Jaroslav Krizek, Oct 27 2013

Keywords

• nonn

Comments

Dirichlet convolution of A080339(n) with itself, where A080339 = characteristic function of noncomposite numbers (A008578).

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 A033942 (numbers with least 3 prime factors (counted with multiplicity)); a(n) = 1 for n = p^2, p = prime; a(n) = 2 for numbers n from A167171 (A006881 union A000040).

Examples

			For n = 6: a(6) = Sum_(d|6) A080339(d) * A080339(6/d) = 1*0 + 1*1 + 1*1 + 0*1 = 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. A080339, A033942, A167171, A006881, A000040, A230595.

Programs

Mathematica

a[n_] := Switch[FactorInteger[n][[;;, 2]], {2}, 1, {1}, 2, {1, 1}, 2, , 0]; a[1] = 1; Array[a, 100] (* _Amiram Eldar, Aug 29 2023 *)

PARI

A080339(n) = ((1==n)||isprime(n));
A230594(n) = sumdiv(n,d,(A080339(d)*A080339(n/d))); \\ Antti Karttunen, Jul 27 2017

Python

from sympy import factorint
def A230594(n): return 0 if sum(f:=factorint(n).values())>2 else (1 if n==1 or max(f)==2 else 2) # Chai Wah Wu, Jul 23 2024

Formula

a(n) = Sum_{d|n} A080339(d) * A080339(n/d).