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

1,4

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).

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A132952.

Sequence in context: A163407 A023891 A075091 * A195207 A157721 A085968

Adjacent sequences: A132950 A132951 A132952 * A132954 A132955 A132956

Keyword

nonn

Author

Leroy Quet, Sep 05 2007

Extensions

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

Status

approved