A133828 a(n) = the smallest "isolated divisor" of n, or 0 if no such divisor exists. A positive divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.
1, 0, 1, 4, 1, 6, 1, 4, 1, 5, 1, 6, 1, 7, 1, 4, 1, 6, 1, 10, 1, 11, 1, 6, 1, 13, 1, 4, 1, 10, 1, 4, 1, 17, 1, 6, 1, 19, 1, 8, 1, 14, 1, 4, 1, 23, 1, 6, 1, 5, 1, 4, 1, 6, 1, 4, 1, 29, 1, 10, 1, 31, 1, 4, 1, 6, 1, 4, 1, 5, 1, 6, 1, 37, 1, 4, 1, 6, 1, 8, 1, 41, 1, 12, 1, 43, 1, 4, 1, 15, 1, 4, 1, 47, 1, 6, 1
Offset: 1
Keywords
Examples
a(18)=6 because the isolated divisors of 18 are 6,9 and 18.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..16384
- Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Programs
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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(a(j)[1],j=3..80); # Emeric Deutsch, Oct 16 2007 A133828 := proc(n) local divs,k,i ; divs := sort(convert(numtheory[divisors](n),list)) ; for i from 1 to nops(divs) do k := op(i,divs) ; if not k-1 in divs and not k+1 in divs then RETURN(k) ; fi ; od: RETURN(0) ; end: seq(A133828(n),n=1..100) ; # R. J. Mathar, Oct 19 2007 -
Mathematica
a[n_] := If[OddQ[n], 1, For[d = 2, d <= n, d++, If[Divisible[n, d] && !Divisible[n, d-1] && !Divisible[n, d+1], Return[d]]]] /. Null -> 0; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jul 20 2024 *) -
PARI
A133828(n) = if(n%2,1,fordiv(n,d,if((d>1)&&(n%(d-1))&&(n%(d+1)), return(d))); (0)); \\ Antti Karttunen, Apr 01 2021
Extensions
More terms from Emeric Deutsch and R. J. Mathar, Oct 16 2007
Comments