OFFSET
1,3
COMMENTS
Number of primitive roots in the field with p elements.
Kátai proves that phi(p-1)/(p-1) has a continuous distribution function. - Charles R Greathouse IV, Jul 15 2013
For odd primes p, phi(p-1)<=(p-1)/2 since p has phi(p-1) primitive roots and (p-1)/2 quadratic residues and no primitive root is a quadratic residue. - Geoffrey Critzer, Apr 18 2015
REFERENCES
D. H. Lehmer and Emma Lehmer, "Heuristics Anyone?", in: G. Szegö et al. (eds.), Studies in Mathematical Analysis and Related Topics: Essays in Honor of George Pólya, Stanford University Press, 1962, pp. 202-210.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
P. Erdős, On the density of some sequences of numbers, III., J. London Math. Soc. 13 (1938), pp. 119-127.
Imre Kátai, On distribution of arithmetical functions on the set prime plus one, Compositio Math. 19 (1968), pp. 278-289.
S. S. Pillai, On the sum function connected with primitive roots, Proceedings of the Indian Academy of Sciences - Section A, Vol. 13. No. 6 (1941), 526-529; alternative link.
I. J. Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Mathematische Zeitschrift 28:1 (1928), pp. 171-199.
P. J. Stephens, An average result for Artin's conjecture, Mathematika, Vol. 16, No. 2 (1969), pp. 178-188.
FORMULA
a(n) = phi(phi(prime(n))). - Robert G. Wilson v, Dec 26 2015
a(n) = phi(A006093(n)). - Michel Marcus, Dec 27 2015
Sum_{k; prime(k) <= x} a(k)/(prime(k)-1) = A * li(x) + O(x/log(x)^D), where A is Artin's constant (A005596), li(x) is the logarithmic integral, and D > 1 (Pillai, 1941; Lehmer and Lehmer 1962; Stephens, 1969). - Amiram Eldar, Jul 23 2025
MATHEMATICA
Table[ EulerPhi[ Prime@n - 1], {n, 70}] (* Robert G. Wilson v, Dec 17 2005 *)
PROG
(PARI) a(n)=eulerphi(prime(n)-1) \\ Charles R Greathouse IV, Dec 08 2011
(Magma) [EulerPhi(NthPrime(n)-1): n in [1..80]]; // Vincenzo Librandi, Apr 06 2015
CROSSREFS
KEYWORD
nonn,look
AUTHOR
STATUS
approved