A339558 Number of divisors of 2n that are the average of a pair of twin primes.

0, 1, 1, 1, 0, 3, 0, 1, 2, 1, 0, 3, 0, 1, 2, 1, 0, 4, 0, 1, 2, 1, 0, 3, 0, 1, 2, 1, 0, 5, 0, 1, 1, 1, 0, 5, 0, 1, 1, 1, 0, 4, 0, 1, 3, 1, 0, 3, 0, 1, 2, 1, 0, 5, 0, 1, 1, 1, 0, 5, 0, 1, 3, 1, 0, 3, 0, 1, 2, 1, 0, 5, 0, 1, 3, 1, 0, 3, 0, 1, 2, 1, 0, 4, 0, 1, 1, 1, 0, 7, 0

Offset: 1

Author

Wesley Ivan Hurt, Dec 08 2020

Keywords

• easy
• nonn

Examples

			a(6) = 3; There are 3 divisors of 2*6 = 12 that are the average of twin primes, namely 4, 6 and 12.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A010051, A014574, A241560, A284203.

Programs

Maple

f:= proc(n) nops(select(t -> isprime(t-1) and isprime(t+1), numtheory:-divisors(2*n))) end proc:
map(f, [$1..100]); # Robert Israel, Jan 06 2021

Mathematica

Table[Sum[(PrimePi[2n/i + 1] - PrimePi[2n/i]) (PrimePi[2n/i - 1] - PrimePi[2n/i - 2]) (1 - Ceiling[2n/i] + Floor[2n/i]), {i, 2n}], {n, 100}]

PARI

a(n) = sumdiv(2*n, d, (d>1) && (bigomega(d^2-1)==2)); \\ Michel Marcus, Dec 16 2020

PARI

a(n) = sumdiv(2*n, d, d > 1 && isprime(d-1) && isprime(d+1)); \\ Amiram Eldar, Jun 03 2024

Formula

a(n) = Sum_{d|(2*n)} c(d+1) * c(d-1), where c is the prime characteristic (A010051).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 * A241560 = 1.857671... . - Amiram Eldar, Jun 03 2024