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A132881
a(n) is the number of isolated divisors of n.
27
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
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
Sequence in context: A178771 A289498 A193929 * A224702 A267263 A060130
KEYWORD
nonn
AUTHOR
Leroy Quet, Sep 03 2007
EXTENSIONS
More terms from Olivier Gérard, Sep 22 2007
STATUS
approved