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%I #52 Sep 08 2022 08:44:35
%S 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,
%T 32,52,36,48,36,48,64,44,72,40,48,54,82,84,88,48,72,64,84,60,48,72,
%U 112,72,112,96,64,100,128,130,132,72,88,96,92,144,96,120,96,156,80,96,172,112
%N phi(p-1), as p runs through the primes.
%C Number of primitive roots in the field with p elements.
%C Kátai proves that phi(p-1)/(p-1) has a continuous distribution function. - _Charles R Greathouse IV_, Jul 15 2013
%C 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
%H T. D. Noe, <a href="/A008330/b008330.txt">Table of n, a(n) for n = 1..10000</a>
%H P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1938-09.pdf">On the density of some sequences of numbers, III.</a>, J. London Math. Soc. 13 (1938), pp. 119-127.
%H Imre Kátai, <a href="http://archive.numdam.org/article/CM_1968__19_4_278_0.pdf">On distribution of arithmetical functions on the set prime plus one</a>, Compositio Math. 19 (1968), pp. 278-289.
%H I. J. Schoenberg, <a href="http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002370158">Über die asymptotische Verteilung reeller Zahlen mod 1</a>, Mathematische Zeitschrift 28:1 (1928), pp. 171-199.
%F a(n) = phi(phi(prime(n))). - _Robert G. Wilson v_, Dec 26 2015
%F a(n) = phi(A006093(n)). - _Michel Marcus_, Dec 27 2015
%p A008330 := proc(n)
%p numtheory[phi](ithprime(n)-1) ;
%p end proc:
%p seq(A008330(n),n=1..100) ;
%t Table[ EulerPhi[ Prime@n - 1], {n, 70}] (* _Robert G. Wilson v_, Dec 17 2005 *)
%o (PARI) a(n)=eulerphi(prime(n)-1) \\ _Charles R Greathouse IV_, Dec 08 2011
%o (Magma) [EulerPhi(NthPrime(n)-1): n in [1..80]]; // _Vincenzo Librandi_, Apr 06 2015
%Y Cf. A000010, A241194, A241195 (fraction phi(p-1)/(p-1)), A338364 (partial products).
%K nonn,look
%O 1,3
%A _N. J. A. Sloane_
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