A132953 a(n) is the sum of the isolated totatives of n.

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, 168, 0, 120, 0, 256, 33, 272, 35, 216, 0, 342, 39, 320, 0, 252, 0, 440, 135, 506, 0, 384, 0, 500, 51, 624, 0, 486, 55, 672, 57, 812, 0, 480, 0, 930, 189, 1024, 65, 660, 0, 1088, 69

Offset: 1

Author

Leroy Quet, Sep 05 2007

Keywords

• nonn

Comments

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.

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

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A132952.

Programs

Mathematica

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 *)

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

Formula

a(n) = (n/2) * A132952(n). - Robert G. Wilson v, Sep 13 2007

Extensions

Edited and extended by Robert G. Wilson v, Sep 13 2007