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

%I A332085 #24 May 30 2025 15:48:03
%S A332085 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,
%T A332085 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,
%U A332085 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
%N A332085 Number of ordered pairs of divisors of n, (d1,d2), such that d1 is prime and d1 <= d2.
%F A332085 a(n) = Sum_{d1|n, d2|n, d1 is prime, d1 <= d2} 1.
%F A332085 a(n) = A337320(n) + omega(n).
%F A332085 a(n) = Sum_{p|n, p prime} A135539(n,p). - _Ridouane Oudra_, May 24 2025
%F A332085 a(n) = A248577(n) - A337322(n). - _Ridouane Oudra_, May 30 2025
%e A332085 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.
%e A332085 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.
%e A332085 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.
%e A332085 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.
%t A332085 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}]
%o A332085 (PARI) row(n) = my(d=divisors(n)); vector(n, k, #select(x->(x>=k), d)); \\ A135539
%o A332085 a(n) = my(v=row(n)); sumdiv(n, d, if (isprime(d), v[d])); \\ _Michel Marcus_, May 24 2025
%Y A332085 Cf. A001221 (omega), A135539, A337228, A337320, A337322, A248577.
%K A332085 nonn
%O A332085 1,4
%A A332085 _Wesley Ivan Hurt_, Aug 22 2020