A332085 Number of ordered pairs of divisors of n, (d1,d2), such that d1 is prime and d1 <= d2.

0, 1, 1, 2, 1, 5, 1, 3, 2, 5, 1, 9, 1, 5, 5, 4, 1, 9, 1, 8, 5, 5, 1, 13, 2, 5, 3, 8, 1, 18, 1, 5, 5, 5, 5, 15, 1, 5, 5, 12, 1, 17, 1, 8, 9, 5, 1, 17, 2, 9, 5, 8, 1, 13, 5, 12, 5, 5, 1, 29, 1, 5, 9, 6, 5, 17, 1, 8, 5, 18, 1, 21, 1, 5, 9, 8, 5, 17, 1, 16, 4, 5, 1, 28, 5, 5, 5, 11, 1, 30

Offset: 1

Author

Wesley Ivan Hurt, Aug 22 2020

Keywords

• nonn

Examples

			a(7) = 1; There are two divisors of 7: {1,7}. If we list the ordered pairs of divisors of n, (d1,d2) where d1 is prime and d1 <= d2, we get (7,7). So a(7) = 1.
a(8) = 3; There are 4 divisors of 8: {1,2,4,8}. If we list the ordered pairs of divisors of n, (d1,d2) where d1 is prime and d1 <= d2, we get (2,2), (2,4) and (2,8). So a(8) = 3.
a(9) = 2; There are three divisors of 9: {1,3,9}. If we list the ordered pairs of divisors of n, (d1,d2) where d1 is prime and d1 <= d2, we get (3,3) and (3,9). So a(9) = 2.
a(10) = 5; There are four divisors of 10: {1,2,5,10}. If we list the ordered pairs of divisors of n, (d1,d2) where d1 is prime and d1 <= d2, we get (2,2), (2,5), (2,10), (5,5) and (5,10). So a(10) = 5.
		

Crossrefs

Cf. A001221 (omega), A135539, A337228, A337320, A337322, A248577.

Programs

Mathematica

Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k}], {k, n}], {n, 100}]

PARI

row(n) = my(d=divisors(n)); vector(n, k, #select(x->(x>=k), d)); \\ A135539
a(n) = my(v=row(n)); sumdiv(n, d, if (isprime(d), v[d])); \\ Michel Marcus, May 24 2025

Formula

a(n) = Sum_{d1|n, d2|n, d1 is prime, d1 <= d2} 1.

a(n) = A337320(n) + omega(n).

a(n) = Sum_{p|n, p prime} A135539(n,p). - Ridouane Oudra, May 24 2025

a(n) = A248577(n) - A337322(n). - Ridouane Oudra, May 30 2025