A132747 a(n) = number of non-isolated divisors of n.

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

Offset: 1

Author

Leroy Quet, Aug 27 2007

Keywords

• nonn

Comments

A divisor d of n is non-isolated if either d-1 or d+1 divides n. a(2n-1) = 0 for all n >= 1.

Examples

			The positive divisors of 20 are 1,2,4,5,10,20. Of these, 1 and 2 are next to each other and 4 and 5 are next to each other. So a(20) = the number of these divisors, which is 4.
		

Links

• Ray Chandler, Table of n, a(n) for n=1..10000

Crossrefs

Cf. A000005, A002162, A129308, A132748, A132881.

Programs

Mathematica

Table[Length[Select[Divisors[n], If[ # > 1, IntegerQ[n/(#*(# - 1))]] || IntegerQ[n/(#*(# + 1))] &]], {n, 1, 90}] (* Stefan Steinerberger, Oct 26 2007 *)

PARI

a(n) = my(div = divisors(n)); sumdiv(n, d, vecsearch(div, d-1) || vecsearch(div, d+1)); \\ Michel Marcus, Aug 22 2014

Formula

a(n) = A000005(n) - A132881(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(2) + 1 = A002162 + 1 = 1.693147.... . - Amiram Eldar, Mar 22 2024

Extensions

More terms from Stefan Steinerberger, Oct 26 2007

Extended by Ray Chandler, Jun 24 2008