A243865 Number of twin divisors of n.

0, 0, 2, 2, 0, 2, 0, 2, 2, 0, 0, 5, 0, 0, 3, 2, 0, 2, 0, 2, 2, 0, 0, 6, 0, 0, 2, 2, 0, 3, 0, 2, 2, 0, 2, 5, 0, 0, 2, 4, 0, 2, 0, 2, 3, 0, 0, 6, 0, 0, 2, 2, 0, 2, 0, 2, 2, 0, 0, 8, 0, 0, 4, 2, 0, 2, 0, 2, 2, 2, 0, 6, 0, 0, 3, 2, 0, 2, 0, 4, 2, 0, 0, 7, 0, 0, 2, 2, 0, 3, 0, 2, 2, 0, 0, 6

Offset: 1

Author

Juri-Stepan Gerasimov, Jun 13 2014

Keywords

• nonn

Comments

A divisor m of n is a twin divisor if m-2 (for m >= 3) and m+2 (for m <= n-2) also divide n.

Examples

			The positive divisors of 20 are 1, 2, 4, 5, 10, 20. Of these, 2 and 4 are twin divisors: (2)+2 = 4, which divides n, and (4)-2 = 2 also divides n. So a(20) = the number of these divisors, which is 2.
		

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A002162, A099475, A132747, A243917.

Programs

PARI

a(n) = sumdiv(n, d, ((d>2) && !(n % (d-2))) || !(n % (d+2))); \\ Michel Marcus, Jun 25 2014

Formula

a(n) = A000005(n) - A243917(n).

a(3n) > 1 for all n >= 1.

a(A099477(n)) = 0, a(A059267(n)) > 0.

A099475(n) <= a(n) <= A000005(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(2)/2 + 17/12 = 1.7632402569... . - Amiram Eldar, Mar 22 2024