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

1,2

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.

Links

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

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A324848 A090330 A332447 * A301979 A183063 A318979

Adjacent sequences: A132744 A132745 A132746 * A132748 A132749 A132750

Keyword

nonn

Author

Leroy Quet, Aug 27 2007

Extensions

More terms from Stefan Steinerberger, Oct 26 2007

Extended by Ray Chandler, Jun 24 2008

Status

approved