A070554 Number of positive integers, k, where k <= 2n+1 and gcd(k, 2n+1) = gcd(k+1, 2n+1) = 1.

1, 1, 3, 5, 3, 9, 11, 3, 15, 17, 5, 21, 15, 9, 27, 29, 9, 15, 35, 11, 39, 41, 9, 45, 35, 15, 51, 27, 17, 57, 59, 15, 33, 65, 21, 69, 71, 15, 45, 77, 27, 81, 45, 27, 87, 55, 29, 51, 95, 27, 99, 101, 15, 105, 107, 35, 111, 63, 33, 75, 99, 39, 75, 125, 41, 129, 85, 27, 135, 137

Offset: 0

Author

Leroy Quet, Nov 15 2000

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• Walter Klotz and Torsten Sander, Some Properties of Unitary Cayley Graphs, The Electronic Journal of Combinatorics, Volume 14 (2007), #R45. See Corollary 7, p. 4.

Crossrefs

Bisection of A058026.

Cf. A065474.

Programs

Maple

A070554:=proc(n) local p, a:=2*n+1; for p in numtheory[factorset](2*n+1) do a:=a*(1-2/p) end do; a end proc: seq(A070554(n), n=0..100); # Ridouane Oudra, Aug 20 2024

Mathematica

f[p_, e_] := (p-2) * p^(e-1); a[0] = 1; a[n_] := Times @@ f @@@ FactorInteger[2*n+1]; Array[a, 100, 0] (* Amiram Eldar, Jun 22 2025 *)

PARI

a(n) = my(n = 2*n+1); n*prod(p=1, n, if (isprime(p) && !(n % p), (1-2/p), 1)); \\ Michel Marcus, Feb 02 2016

PARI

a(n) = {my(f = factor(2*n+1)); prod(i=1, #f~, (f[i,1]-2) * f[i,1]^(f[i,2]-1));} \\ Amiram Eldar, Jun 22 2025

Formula

a(n) = A058026(2*n+1). - Ridouane Oudra, Aug 20 2024

Sum_{k=0..n} a(k) ~ c * n^2, where c = 2 * A065474. - Amiram Eldar, Jun 22 2025

Extensions

More terms from Sascha Kurz, Feb 02 2003