A008330 phi(p-1), as p runs through the primes.

1, 1, 2, 2, 4, 4, 8, 6, 10, 12, 8, 12, 16, 12, 22, 24, 28, 16, 20, 24, 24, 24, 40, 40, 32, 40, 32, 52, 36, 48, 36, 48, 64, 44, 72, 40, 48, 54, 82, 84, 88, 48, 72, 64, 84, 60, 48, 72, 112, 72, 112, 96, 64, 100, 128, 130, 132, 72, 88, 96, 92, 144, 96, 120, 96, 156, 80, 96, 172, 112

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• look

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.

Crossrefs

Cf. A000010, A005596, A241194, A241195 (fraction phi(p-1)/(p-1)), A338364 (partial products).

Programs

Magma

[EulerPhi(NthPrime(n)-1): n in [1..80]]; // Vincenzo Librandi, Apr 06 2015

Maple

A008330 := proc(n)
    numtheory[phi](ithprime(n)-1) ;
end proc:
seq(A008330(n),n=1..100) ;

Mathematica

Table[ EulerPhi[ Prime@n - 1], {n, 70}] (* Robert G. Wilson v, Dec 17 2005 *)

PARI

a(n)=eulerphi(prime(n)-1) \\ Charles R Greathouse IV, Dec 08 2011

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