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

1,3

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

Links

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

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A132882, A132747.

Sequence in context: A178771 A289498 A193929 * A224702 A267263 A060130

Adjacent sequences: A132878 A132879 A132880 * A132882 A132883 A132884

Keyword

nonn

Author

Leroy Quet, Sep 03 2007

Extensions

More terms from Olivier Gérard, Sep 22 2007

Status

approved