A132881 a(n) is the number of isolated divisors of n.

1, 0, 2, 1, 2, 1, 2, 2, 3, 2, 2, 2, 2, 2, 4, 3, 2, 3, 2, 2, 4, 2, 2, 4, 3, 2, 4, 4, 2, 3, 2, 4, 4, 2, 4, 5, 2, 2, 4, 4, 2, 3, 2, 4, 6, 2, 2, 6, 3, 4, 4, 4, 2, 5, 4, 4, 4, 2, 2, 6, 2, 2, 6, 5, 4, 5, 2, 4, 4, 6, 2, 6, 2, 2, 6, 4, 4, 5, 2, 6, 5, 2, 2, 6, 4, 2, 4, 6, 2, 5, 4, 4, 4, 2, 4, 8, 2, 4, 6, 5, 2, 5, 2, 6, 8

Offset: 1

Author

Leroy Quet, Sep 03 2007

Keywords

• nonn

Comments

A divisor d of n is isolated if neither d-1 nor d+1 divides n.

The convention for 1 is that it is an isolated divisor iff n is odd. - Olivier Gérard, Sep 22 2007

Examples

			The positive divisors of 56 are 1,2,4,7,8,14,28,56. Of these, 1 and 2 are adjacent and 7 and 8 are adjacent. The isolated divisors are therefore 4,14,28,56. There are 4 of these, so a(56) = 4.
		

Links

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

Crossrefs

Cf. A132882, A132747.

Programs

Maple

with(numtheory): a:=proc(n) local div, ISO, i: div:=divisors(n): ISO:={}: for i to tau(n) do if member(div[i]-1, div)=false and member(div[i]+1, div)=false then ISO:=`union`(ISO,{div[i]}) end if end do end proc; 1, 0, seq(nops(a(j)), j=3..105); # Emeric Deutsch, Oct 02 2007

Mathematica

Table[Length@Select[Divisors[n],(#==1||Mod[n,#-1]>0)&&Mod[n,#+1]>0&],{n,1,200}] (* Olivier Gérard Sep 22 2007 *)
id[n_]:=DivisorSigma[0,n]-Length[Union[Flatten[Select[Partition[Divisors[ n],2,1],#[[2]]-#[[1]]==1&]]]]; Array[id,110] (* Harvey P. Dale, Jun 04 2018 *)

Formula

a(n) = A000005(n) - A132747(n).

Extensions

More terms from Olivier Gérard, Sep 22 2007