A132953 - OEIS

%I #18 Nov 01 2018 18:20:13

%S 0,1,0,4,0,6,0,16,0,20,0,24,0,42,15,64,0,54,0,80,21,110,0,96,0,156,0,

%T 168,0,120,0,256,33,272,35,216,0,342,39,320,0,252,0,440,135,506,0,384,

%U 0,500,51,624,0,486,55,672,57,812,0,480,0,930,189,1024,65,660,0,1088,69

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

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

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

%H Antti Karttunen, <a href="/A132953/b132953.txt">Table of n, a(n) for n = 1..16384</a>

%H Antti Karttunen, <a href="/A132953/a132953.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%F a(n) = (n/2) * A132952(n). - _Robert G. Wilson v_, Sep 13 2007

%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. Therefore a(15) = 4 + 11 = 15.

%t fQ[k_, n_] := GCD[k, n] == 1 && GCD[k - 1, n] > 1 && GCD[k + 1, n] > 1; f[n_] := Plus @@ Select[ Rest[ Range@n - 1], fQ[ #, n] &]; Array[f, 69] (* _Robert G. Wilson v_ *)

%o (PARI) A132953(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 += k); pg = g; g = ng); (s); }; \\ _Antti Karttunen_, Nov 01 2018

%Y Cf. A132952.

%K nonn

%O 1,4

%A _Leroy Quet_, Sep 05 2007

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