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A339558
Number of divisors of 2n that are the average of a pair of twin primes.
1
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
1,6
LINKS
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
EXAMPLE
a(6) = 3; There are 3 divisors of 2*6 = 12 that are the average of twin primes, namely 4, 6 and 12.
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}]
PROG
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Wesley Ivan Hurt, Dec 08 2020
STATUS
approved