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%I #19 Jul 20 2024 08:09:47
%S 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,
%T 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,
%U 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
%N 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.
%C a(2n-1) = 1 for all positive integers n. 2 has no isolated divisors. a(2) is 0 only as a placeholder.
%H Antti Karttunen, <a href="/A133828/b133828.txt">Table of n, a(n) for n = 1..16384</a>
%H Antti Karttunen, <a href="/A133828/a133828.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%e a(18)=6 because the isolated divisors of 18 are 6,9 and 18.
%p 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
%p 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
%t 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;
%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jul 20 2024 *)
%o (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
%Y Cf. A133779, A133829, A134187, A134188.
%K nonn
%O 1,4
%A _Leroy Quet_, Sep 25 2007
%E More terms from _Emeric Deutsch_ and _R. J. Mathar_, Oct 16 2007