A339021 - OEIS

%I #11 Nov 26 2020 22:02:39

%S 1,3,6,6,4,15,5,9,13,13,5,26,3,11,21,12,3,30,3,22,23,12,3,40,9,10,22,

%T 20,3,52,3,14,23,11,19,51,4,8,21,32,5,55,3,20,47,7,1,53,10,27,18,16,4,

%U 49,12,32,15,8,2,92,1,6,48,17,11,55,2,20,16,46,2,79,3,10,42,18,14,53,4,42

%N Number of pairs of divisors of 2n, (d1,d2), such that d1 < d2 and at least one of d1*d2 +- 1 is prime.

%C a(n) >= 1 since for all n >= 1, 2n has the divisor pair (1,2) with 1 < 2 and 1*2 + 1 = 3 (prime).

%F a(n) = Sum_{d1|(2*n), d2|(2*n), d1 < d2} sign(c(d1*d2 + 1) + c(d1*d2 - 1)), where c is the prime characteristic (A010051).

%e a(3) = 6; 2*3 = 6 has 6 divisor pairs (d1,d2) such that d1 < d2 where at least one of d1*d2 +- 1 is prime: (1,2), (1,3), (1,6), (2,3), (2,6), (3,6) as (1*2 + 1) = 3, (1*3 - 1) = 2, (1*6 + 1) = 7, (2*3 - 1) = 5, (2*6 - 1) = 11, and (3*6 - 1) = 17 (all prime).

%t Table[Sum[Sum[Sign[PrimePi[i*k - 1] - PrimePi[i*k - 2] + PrimePi[i*k + 1] -  PrimePi[i*k]]*(1 - Ceiling[2 n/k] + Floor[2 n/k]) (1 - Ceiling[2 n/i] + Floor[2 n/i]), {i, k - 1}], {k, 2 n}], {n, 80}]

%Y Cf. A010051, A337982.

%K nonn

%O 1,2

%A _Wesley Ivan Hurt_, Nov 22 2020