A339558 - OEIS

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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

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OFFSET

1,6

LINKS

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

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

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

Sequence in context: A242869 A224878 A129555 * A147755 A307195 A231188

Adjacent sequences: A339555 A339556 A339557 * A339559 A339560 A339561

KEYWORD

easy,nonn

AUTHOR

Wesley Ivan Hurt, Dec 08 2020

STATUS

approved