1,3

Number of elements in a reduced residue system modulo n.

Degree of the n-th cyclotomic polynomial (cf. A013595). - Benoit Cloitre, Oct 12 2002

Number of distinct generators of a cyclic group of order n. Number of primitive n-th roots of unity. (A primitive n-th root x is such that x^k is not equal to 1 for k = 1, 2, ..., n - 1, but x^n = 1.) - Lekraj Beedassy, Mar 31 2005

Also number of complex Dirichlet characters modulo n; Sum_{k=1..n} a(k) is asymptotic to (3/Pi^2)*n^2. - Steven Finch, Feb 16 2006

a(n) is the highest degree of irreducible polynomial dividing 1 + x + x^2 + ... + x^(n-1) = (x^n - 1)/(x - 1). - Alexander Adamchuk, Sep 02 2006, corrected Sep 27 2006

a(p) = p - 1 for prime p. a(n) is even for n > 2. For n > 2, a(n)/2 = A023022(n) = number of partitions of n into 2 ordered relatively prime parts. - Alexander Adamchuk, Jan 25 2007

Number of automorphisms of the cyclic group of order n. - Benoit Jubin, Aug 09 2008

a(n+2) equals the number of palindromic Sturmian words of length n which are 'bispecial', prefix or suffix of two Sturmian words of length n + 1. - Fred Lunnon, Sep 05 2010

Suppose that a and n are coprime positive integers, then by Euler's totient theorem, any factor of n divides a^phi(n) - 1. - Lei Zhou, Feb 28 2012

a(n) = A096396(n) + A096397(n). - Reinhard Zumkeller, Mar 24 2012

If m has k prime factors, (p_1, p_2, ..., p_k), then phi(m*n) = (Product_{i=1..k} phi (p_i*n))/phi(n)^(k-1). For example, phi(42*n) = phi(2*n)*phi(3*n)*phi(7*n)/phi(n)^2. - Gary Detlefs, Apr 21 2012

Sum_{n>=1} a(n)/n! = 1.954085357876006213144... This sum is referenced in Plouffe's inverter. - Alexander R. Povolotsky, Feb 02 2013 (see A336334. - Hugo Pfoertner, Jul 22 2020)

The order of the multiplicative group of units modulo n. - Michael Somos, Aug 27 2013

A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - Michael Somos, Dec 30 2016

From Eric Desbiaux, Jan 01 2017: (Start)

a(n) equals the Ramanujan sum c_n(n) (last term on n-th row of triangle A054533).

a(n) equals the Jordan function J_1(n) (cf. A007434, A059376, A059377, which are the Jordan functions J_2, J_3, J_4, respectively). (End)

For n>1, a(n) appears to be equal to the number of semi-meander solutions for n with top arches containing exactly 2 mountain ranges and exactly 2 arches of length 1. - Roger Ford, Oct 11 2017

a(n) is the minimum dimension of a lattice able to generate, via cut-and-project, the quasilattice whose diffraction pattern features n-fold rotational symmetry. The case n=15 is the first n > 1 in which the following simpler definition fails: "a(n) is the minimum dimension of a lattice with n-fold rotational symmetry". - Felix Flicker, Nov 08 2017

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.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 24.

M. Baake and U. Grimm, Aperiodic Order Vol. 1: A Mathematical Invitation, Encyclopedia of Mathematics and its Applications 149, Cambridge University Press, 2013: see Tables 3.1 and 3.2.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 193.

C. W. Curtis, Pioneers of Representation Theory ..., Amer. Math. Soc., 1999; see p. 3.

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, Chapter V.

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 115-119.

Carl Friedrich Gauss, "Disquisitiones Arithmeticae", Yale University Press, 1965; see p. 21.

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2n-d ed.; Addison-Wesley, 1994, p. 137.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 60, 62, 63, 288, 323, 328, 330.

Peter Hilton and Jean Pedersen, A Mathematical Tapestry, Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, pages 261-264, the Coach theorem.

P. Ribenboim, The New Book of Prime Number Records.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Forgues, Table of n, phi(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972.

D. Alpern, Factorization using the Elliptic Curve Method(along with sigma_0, sigma_1 and phi functions)

Joerg Arndt, Matters Computational (The Fxtbook), section 39.7, pp. 776-778.

F. Bayart, Indicateur d'Euler (in French).

A. Bogomolny, Euler Function and Theorem.

C. K. Caldwell, The Prime Glossary, Euler's phi function

R. D. Carmichael, A table of the values of m corresponding to given values of phi(m), Amer. J. Math., 30 (1908),394-400. [Annotated 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]

K. Ford, The number of solutions of phi(x)=m, arXiv:math/9907204 [math.NT], 1999.

Kevin Ford, Florian Luca and Pieter Moree, Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields, arXiv:1108.3805 [math.NT], 2011.

H. Fripertinger, The Euler phi function.

Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.

E. Pérez Herrero, Totient Carnival partitions, Psychedelic Geometry Blogspot

M. Lal and P. Gillard, Table of Euler's phi function, n < 10^5, Math. Comp., 23 (1969), 682-683.

D. N. Lehmer, Review of Dickson's History of the Theory of Numbers, Bull. Amer. Math. Soc., 26 (1919), 125-132.

Peter Luschny, Sequences related to Euler's totient function.

Mathematics Stack Exchange, Is the Euler phi function bounded below? (2013).

Mathforum, Proving phi(m) Is Even.

K. Matthews, Factorizing n and calculating phi(n), omega(n), d(n), sigma(n) and mu(n).

Graeme McRae, Euler's Totient Function.

François Nicolas, A simple, polynomial-time algorithm for the matrix torsion problem, arXiv:0806.2068 [cs.DM], 2009.

Matthew Parker, The first 5 million terms (7-Zip compressed file).

Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp., to appear (2014).

Primefan, Euler's Totient Function Values For n=1 to 500, with Divisor Lists.

Marko Riedel, Combinatorics and number theory page.

J. Barkley Rosser, Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), no. 1, 64-94.

K. Schneider, Euler phi-function, PlanetMath.org.

W. Sierpiński, Euler's Totient Function And The Theorem Of Euler.

U. Sondermann, Euler's Totient Function.

W. A. Stein, Phi is a Multiplicative Function

Pinthira Tangsupphathawat, Takao Komatsu, Vichian Laohakosol, Minimal Polynomials of Algebraic Cosine Values, II, J. Int. Seq., Vol. 21 (2018), Article 18.9.5.

G. Villemin, Totient d'Euler.

A. de Vries, The prime factors of an integer (along with Euler's phi and Carmichael's lambda functions)

K. W. Wegner, Values of phi(x) = n for n from 2 through 1978, mimeographed manuscript, no date. [Annotated scanned copy]

Eric Weisstein's World of Mathematics, Modulo Multiplication Group.

Eric Weisstein's World of Mathematics, Moebius Transform.

Eric Weisstein's World of Mathematics, Totient Function.

Wikipedia, Euler's totient function.

Wikipedia, Multiplicative group of integers modulo n.

Wikipedia, Ramanujan's sum

Wolfram Research, First 50 values of phi(n).

G. Xiao, Numerical Calculator, To display phi(n) operate on "eulerphi(n)".

Index entries for "core" sequences

Index to divisibility sequences

phi(n) = n*Product_{distinct primes p dividing n} (1 - 1/p).

Sum_{d divides n} phi(d) = n.

phi(n) = Sum_{d divides n} mu(d)*n/d, i.e., the Moebius transform of the natural numbers; mu() = Moebius function A008683().

Dirichlet generating function Sum_{n>=1} phi(n)/n^s = zeta(s-1)/zeta(s). Also Sum_{n >= 1} phi(n)*x^n/(1 - x^n) = x/(1 - x)^2.

Multiplicative with a(p^e) = (p - 1)*p^(e-1). - David W. Wilson, Aug 01 2001

Sum_{n>=1} (phi(n)*log(1 - x^n)/n) = -x/(1 - x) for -1 < x < 1 (cf. A002088) - Henry Bottomley, Nov 16 2001

a(n) = binomial(n+1, 2) - Sum_{i=1..n-1} a(i)*floor(n/i) (see A000217 for inverse). - Jon Perry, Mar 02 2004

It is a classical result (certainly known to Landau, 1909) that lim inf n/phi(n) = 1 (taking n to be primes), lim sup n/(phi(n)*log(log(n))) = e^gamma, with gamma = Euler's constant (taking n to be products of consecutive primes starting from 2 and applying Mertens' theorem). See e.g. Ribenboim, pp. 319-320. - Pieter Moree, Sep 10 2004

a(n) = Sum_{i=1..n} |k(n, i)| where k(n, i) is the Kronecker symbol. Also a(n) = n - #{1 <= i <= n : k(n, i) = 0}. - Benoit Cloitre, Aug 06 2004 [Corrected by Jianing Song, Sep 25 2018]

Conjecture: limit Sum_{i>=2} (-1)^i/(i*phi(i)) exists and is approximately 0.558 (A335319). - Orges Leka (oleka(AT)students.uni-mainz.de), Dec 23 2004

From Enrique Pérez Herrero, Sep 07 2010: (Start)

a(n) = Sum_{i=1..n} floor(sigma_k(i*n)/sigma_k(i)*sigma_k(n)), where sigma_2 is A001157.

a(n) = Sum_{i=1..n} floor(tau_k(i*n)/tau_k(i)*tau_k(n)), where tau_3 is A007425.

a(n) = Sum_{i=1..n} floor(rad(i*n)/rad(i)*rad(n)), where rad is A007947. (End)

a(n) = A173557(n)*A003557(n). - R. J. Mathar, Mar 30 2011

phi(p*n) = phi(n)*(floor(((n + p - 1) mod p)/(p - 1)) + p - 1), for primes p. - Gary Detlefs, Apr 21 2012

For odd n, a(n) = 2*A135303((n-1)/2)*A003558((n-1)/2) or phi(n) = 2*c*k; the Coach theorem of Pedersen et al. Cf. A135303. - Gary W. Adamson, Aug 15 2012

G.f.: Sum_{n>=1} mu(n)*x^n/(1 - x^n)^2, where mu(n) = A008683(n). - Mamuka Jibladze, Apr 05 2015

a(n) = n - cototient(n) = n - A051953(n). - Omar E. Pol, May 14 2016

a(n) = lim_{s->1} n*zeta(s)*(Sum_{d divides n} A008683(d)/(e^(1/d))^(s-1)), for n > 1. - Mats Granvik, Jan 26 2017

Conjecture: a(n) = Sum_{a=1..n} Sum_{b=1..n} Sum_{c=1..n} 1 for n > 1. The sum is over a,b,c such that n*c - a*b = 1. - Benedict W. J. Irwin, Apr 03 2017

a(n) = Sum_{j=1..n} gcd(j, n) cos(2*Pi*j/n) = Sum_{j=1..n} gcd(j, n) exp(2*Pi*i*j/n) where i is the imaginary unit. Notice that the Ramanujan's sum c_n(k) := Sum_{j=1..n, gcd(j, n) = 1} exp(2*Pi*i*j*k/n) gives a(n) = Sum_{k|n} k*c_(n/k)(1) = Sum_{k|n} k*mu(n/k). - Michael Somos, May 13 2018

G.f.: x*d/dx(x*d/dx(log(Product_{k>=1} (1 - x^k)^(-mu(k)/k^2)))), where mu(n) = A008683(n). - Mamuka Jibladze, Sep 20 2018

a(n) = Sum_{d|n} A007431(d). - Steven Foster Clark, May 29 2019

G.f. A(x) satisfies: A(x) = x/(1 - x)^2 - Sum_{k>=2} A(x^k). - Ilya Gutkovskiy, Sep 06 2019

a(n) >= sqrt(n/2) (Nicolas). - Hugo Pfoertner, Jun 01 2020

a(n) > n/(exp(gamma)*log(log(n)) + 5/(2*log(log(n)))), except for n=223092870 (Rosser, Schoenfeld). - Hugo Pfoertner, Jun 02 2020

G.f. = x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + 6*x^7 + 4*x^8 + 6*x^9 + 4*x^10 + ...

a(8) = 4 with {1, 3, 5, 7} units modulo 8. a(10) = 4 with {1, 3, 7, 9} units modulo 10. - Michael Somos, Aug 27 2013

with(numtheory): A000010 := phi; [ seq(phi(n), n=1..100) ]; # version 1

with(numtheory): phi := proc(n) local i, t1, t2; t1 := ifactors(n)[2]; t2 := n*mul((1-1/t1[i][1]), i=1..nops(t1)); end; # version 2

Array[EulerPhi, 70]

(Axiom) [eulerPhi(n) for n in 1..100]

(MAGMA) [ EulerPhi(n) : n in [1..100] ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

(PARI) {a(n) = if( n==0, 0, eulerphi(n))}; /* Michael Somos, Feb 05 2011 */

(Sage)

# euler_phi is a standard function in Sage.

def A000010(n): return euler_phi(n)

def A000010_list(n): return [ euler_phi(i) for i in range(1, n+1)]

# Jaap Spies, Jan 07 2007

(PARI) { for (n=1, 100000, write("b000010.txt", n, " ", eulerphi(n))); } \\ Harry J. Smith, Apr 26 2009

(Sage) [euler_phi(n) for n in range(1, 70)]  # Zerinvary Lajos, Jun 06 2009

(Maxima) makelist(totient(n), n, 0, 1000); /* Emanuele Munarini, Mar 26 2011 */

(Haskell) a n = length (filter (==1) (map (gcd n) [1..n])) -- Allan C. Wechsler, Dec 29 2014

(Python)

from sympy.ntheory import totient

print([totient(i) for i in range(1, 70)])  # Indranil Ghosh, Mar 17 2017

Cf. A008683, A003434 (steps to reach 1), A007755, A049108, A002202 (values).

Cf. A005277 (nontotient numbers). For inverse see A002181, A006511, A058277.

Jordan function J_k(n) is a generalization - see A059379 and A059380 (triangle of values of J_k(n)), this sequence (J_1), A007434 (J_2), A059376 (J_3), A059377 (J_4), A059378 (J_5).

Cf. A054521, A023022, A054525, A134540.

Row sums of triangles A134540, A127448, A143239, A143353 and A143276.

Equals right and left borders of triangle A159937. - Gary W. Adamson, Apr 26 2009

Values for prime powers p^e: A006093 (e=1), A036689 (e=2), A135177 (e=3), A138403 (e=4), A138407 (e=5), A138412 (e=6).

Values for perfect powers n^e: A002618 (e=2), A053191 (e=3), A189393 (e=4), A238533 (e=5), A239442 (e=7), A239443 (e=9).

Cf. A003558, A135303.

Cf. A152455, A080737.

Sequence in context: A080737 A152455 A293484 * A003978 A122645 A122646

Adjacent sequences:  A000007 A000008 A000009 * A000011 A000012 A000013

easy,core,nonn,mult,nice,hear

N. J. A. Sloane

approved