A077197 -id:A077197 - cp's OEIS frontend

A010554 a(n) = phi(phi(n)), where phi is the Euler totient function.

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, 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, 8, 12, 16, 16, 8, 20, 16, 20, 8, 24, 8

Offset: 1

N. J. A. Sloane

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

See A046144 for the number of primitive roots mod n. - Wolfdieter Lang, Mar 09 2012

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.
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.

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdos, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
S. R. Finch, Idempotents and Nilpotents Modulo n (arXiv:math.NT/0605019)
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A000010, A049099, A049100, A049107, A077197.

a010554 = a000010 . a000010  -- Reinhard Zumkeller, Dec 26 2012

[EulerPhi(EulerPhi(n)): n in [1..100]]; // Vincenzo Librandi, Feb 24 2018

Maple
```
with(numtheory): f := n->phi(phi(n));
```

Table[EulerPhi[EulerPhi[n]],{n,0,200}] (* Vladimir Joseph Stephan Orlovsky, Nov 10 2009 *)
Nest[EulerPhi[#]&,Range[100],2] (* Harvey P. Dale, Jan 13 2024 *)

a(n)=eulerphi(eulerphi(n)) \\ Charles R Greathouse IV, Feb 06 2017

A290086 a(n) = A289626(A000010(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 5, 2, 5, 5, 8, 2, 4, 5, 5, 3, 6, 5, 8, 5, 4, 5, 11, 5, 8, 8, 8, 8, 13, 5, 11, 4, 13, 8, 21, 5, 11, 8, 13, 6, 12, 8, 11, 8, 17, 13, 19, 4, 21, 13, 11, 11, 15, 8, 21, 8, 11, 17, 21, 8, 18, 17, 18, 13, 19, 13, 28, 11, 21, 11, 21, 13, 19, 17, 10, 21, 22, 13, 25, 11, 28, 21, 45, 13, 28, 18, 21, 12, 28, 17, 49, 11, 21, 21, 27, 17

Offset: 1

Antti Karttunen, Aug 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A010554, A077197, A277906, A289626, A290085, A290087, A290088.

a(n) = A289626(A000010(n)).

A206941 a(n) = phi(lambda(n)), where phi = A000010, lambda = A002322.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 4, 1, 4, 2, 2, 2, 8, 2, 6, 2, 2, 4, 10, 1, 8, 4, 6, 2, 12, 2, 8, 4, 4, 8, 4, 2, 12, 6, 4, 2, 16, 2, 12, 4, 4, 10, 22, 2, 12, 8, 8, 4, 24, 6, 8, 2, 6, 12, 28, 2, 16, 8, 2, 8, 4, 4, 20, 8, 10, 4, 24, 2, 24, 12, 8, 6, 8, 4, 24, 2, 18

Offset: 1

N. J. A. Sloane, Feb 13 2012

nonn

W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 55, Theorem 4.10.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1909-10), 232-238.

Cf. A000010, A002322, A077197.

a206941 = a000010 . a002322 -- Reinhard Zumkeller, Feb 18 2012

Table[EulerPhi@ CarmichaelLambda@ n, {n, 96}] (* Michael De Vlieger, Mar 18 2016 *)

a(n)=eulerphi(lcm(znstar(n)[2])) \\ Charles R Greathouse IV, Feb 21 2013

cp's OEIS Frontend

A010554 a(n) = phi(phi(n)), where phi is the Euler totient function.

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A290086 a(n) = A289626(A000010(n)).

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A206941 a(n) = phi(lambda(n)), where phi = A000010, lambda = A002322.

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Author

Keywords

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Links

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Programs

Haskell

Mathematica

PARI