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%I #51 Jan 13 2024 13:05:48
%S 1,1,1,1,2,1,2,2,2,2,4,2,4,2,4,4,8,2,6,4,4,4,10,4,8,4,6,4,12,4,8,8,8,
%T 8,8,4,12,6,8,8,16,4,12,8,8,10,22,8,12,8,16,8,24,6,16,8,12,12,28,8,16,
%U 8,12,16,16,8,20,16,20,8,24,8
%N a(n) = phi(phi(n)), where phi is the Euler totient function.
%C If n has a primitive root, then it has exactly phi(phi(n)) of them (Burton 1989, p. 188), which means that if p is a prime number, then there are exactly phi(p-1) incongruent primitive roots of p (Burton 1989). - _Jonathan Vos Post_, Sep 10 2010
%C See A046144 for the number of primitive roots mod n. - _Wolfdieter Lang_, Mar 09 2012
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D Burton, D. M. "The Order of an Integer Modulo n," "Primitive Roots for Primes," and "Composite Numbers Having Primitive Roots." Sections 8.1-8.3 in Elementary Number Theory, 4th ed. Dubuque, IA: William C. Brown Publishers, pp. 184-205, 1989.
%H T. D. Noe, <a href="/A010554/b010554.txt">Table of n, a(n) for n = 1..10000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, <a href="http://math.dartmouth.edu/~carlp/iterate.pdf">On the normal behavior of the iterates of some arithmetic functions</a>, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
%H Paul Erdos, Andrew Granville, Carl Pomerance and Claudia Spiro, <a href="/A000010/a000010_1.pdf">On the normal behavior of the iterates of some arithmetic functions</a>, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
%H S. R. Finch, <a href="http://arXiv.org/abs/math.NT/0605019">Idempotents and Nilpotents Modulo n</a> (arXiv:math.NT/0605019)
%H Boris Putievskiy, <a href="https://arxiv.org/abs/1212.2732">Transformations [Of] Integer Sequences And Pairing Functions</a>, arXiv preprint arXiv:1212.2732 [math.CO], 2012.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimitiveRoot.html">Primitive Root.</a>
%p with(numtheory): f := n->phi(phi(n));
%t Table[EulerPhi[EulerPhi[n]],{n,0,200}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 10 2009 *)
%t Nest[EulerPhi[#]&,Range[100],2] (* _Harvey P. Dale_, Jan 13 2024 *)
%o (Haskell)
%o a010554 = a000010 . a000010 -- _Reinhard Zumkeller_, Dec 26 2012
%o (PARI) a(n)=eulerphi(eulerphi(n)) \\ _Charles R Greathouse IV_, Feb 06 2017
%o (Magma) [EulerPhi(EulerPhi(n)): n in [1..100]]; // _Vincenzo Librandi_, Feb 24 2018
%Y Cf. A000010, A049099, A049100, A049107, A077197.
%K nonn,nice
%O 1,5
%A _N. J. A. Sloane_
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