%I #19 Nov 01 2018 18:20:20

%S 0,1,0,2,0,2,0,4,0,4,0,4,0,6,2,8,0,6,0,8,2,10,0,8,0,12,0,12,0,8,0,16,

%T 2,16,2,12,0,18,2,16,0,12,0,20,6,22,0,16,0,20,2,24,0,18,2,24,2,28,0,

%U 16,0,30,6,32,2,20,0,32,2,24,0,24,0,36,10,36,2,24,0,32,0,40,0,24,2,42,2,40,0

%N a(n) is the number of isolated totatives of n.

%C An isolated totative, k, of n is a positive integer which is less than and coprime to n and is such that neither (k-1) nor (k+1) are coprime to n.

%C a(2n) = phi(2n), where phi(n) = A000010(n).

%C If k is an isolated totative so is n-k. - _Robert G. Wilson v_, Sep 13 2007

%C a(n)=0 for n's: A061345 "Odd prime powers". - _Robert G. Wilson v_, Sep 13 2007

%H Antti Karttunen, <a href="/A132952/b132952.txt">Table of n, a(n) for n = 1..16384</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Totative.html">Totative</a>.

%H Antti Karttunen, <a href="/A132952/a132952.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%e The positive integers which are <= 15 and are coprime to 15 are 1,2,4,7,8,11,13,14. Of these, 1 and 2 are adjacent, 7 and 8 are adjacent and 13 and 14 are adjacent. So the isolated totatives of 15 are 4 and 11. There are 2 of these, so a(15) = 2.

%t fQ[k_, n_] := GCD[k, n] == 1 && GCD[k - 1, n] > 1 && GCD[k + 1, n] > 1; f[n_] := Length@ Select[ Rest[ Range@n - 1], fQ[ #, n] &]; Array[f, 89] (* _Robert G. Wilson v_ *)

%o (PARI) A132952(n) = { my(s=0,pg=0,g=1,ng); for(k=1,n-1,if((1!=(ng=gcd(n,k+1)))&&(1==g)&&(1!=pg),s++); pg = g; g = ng); (s); }; \\ _Antti Karttunen_, Nov 01 2018

%Y Cf. A132953.

%K nonn

%O 1,4

%A _Leroy Quet_, Sep 05 2007

%E Edited and extended by _Robert G. Wilson v_, Sep 13 2007