A132952 a(n) is the number of isolated totatives of n.

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, 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, 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

Offset

1,4

Comments

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.

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

If k is an isolated totative so is n-k. - Robert G. Wilson v, Sep 13 2007

a(n)=0 for n's: A061345 "Odd prime powers". - Robert G. Wilson v, Sep 13 2007

Links

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

Eric Weisstein's World of Mathematics, Totative.

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

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. There are 2 of these, so a(15) = 2.

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A132953.

Sequence in context: A346773 A362208 A167156 * A319998 A336120 A029187

Adjacent sequences: A132949 A132950 A132951 * A132953 A132954 A132955

Keyword

nonn

Author

Leroy Quet, Sep 05 2007

Extensions

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

Status

approved