A006511 -id:A006511 - cp's OEIS frontend

A000010 Euler totient function phi(n): count numbers <= n and prime to n.

1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20, 32, 24, 52, 18, 40, 24, 36, 28, 58, 16, 60, 30, 36, 32, 48, 20, 66, 32, 44

Offset: 1

N. J. A. Sloane

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
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
Number of cyclic Latin squares of order n with the first row in ascending order. - Eduard I. Vatutin, Nov 01 2020
a(n) is the number of rational numbers p/q >= 0 (in lowest terms) such that p + q = n. - Rémy Sigrist, Jan 17 2021
From Richard L. Ollerton, May 08 2021: (Start)
Formulas for the numerous OEIS entries involving Dirichlet convolution of a(n) and some sequence h(n) can be derived using the following (n >= 1):
Sum_{d|n} phi(d)*h(n/d) = Sum_{k=1..n} h(gcd(n,k)) [see P. H. van der Kamp link] = Sum_{d|n} h(d)*phi(n/d) = Sum_{k=1..n} h(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). Similarly,
Sum_{d|n} phi(d)*h(d) = Sum_{k=1..n} h(n/gcd(n,k)) = Sum_{k=1..n} h(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)).
More generally,
Sum_{d|n} h(d) = Sum_{k=1..n} h(gcd(n,k))/phi(n/gcd(n,k)) = Sum_{k=1..n} h(n/gcd(n,k))/phi(n/gcd(n,k)).
In particular, for sequences involving the Möbius transform:
Sum_{d|n} mu(d)*h(n/d) = Sum_{k=1..n} h(gcd(n,k))*mu(n/gcd(n,k))/phi(n/gcd(n,k)) = Sum_{k=1..n} h(n/gcd(n,k))*mu(gcd(n,k))/phi(n/gcd(n,k)), where mu = A008683.
Use of gcd(n,k)*lcm(n,k) = n*k and phi(gcd(n,k))*phi(lcm(n,k)) = phi(n)*phi(k) provide further variations. (End)
From Richard L. Ollerton, Nov 07 2021: (Start)
Formulas for products corresponding to the sums above may found using the substitution h(n) = log(f(n)) where f(n) > 0 (for example, cf. formulas for the sum A018804 and product A067911 of gcd(n,k)):
Product_{d|n} f(n/d)^phi(d) = Product_{k=1..n} f(gcd(n,k)) = Product_{d|n} f(d)^phi(n/d) = Product_{k=1..n} f(n/gcd(n,k))^(phi(gcd(n,k))/phi(n/gcd(n,k))),
Product_{d|n} f(d)^phi(d) = Product_{k=1..n} f(n/gcd(n,k)) = Product_{k=1..n} f(gcd(n,k))^(phi(gcd(n,k))/phi(n/gcd(n,k))),
Product_{d|n} f(d) = Product_{k=1..n} f(gcd(n,k))^(1/phi(n/gcd(n,k))) = Product_{k=1..n} f(n/gcd(n,k))^(1/phi(n/gcd(n,k))),
Product_{d|n} f(n/d)^mu(d) = Product_{k=1..n} f(gcd(n,k))^(mu(n/gcd(n,k))/phi(n/gcd(n,k))) = Product_{k=1..n} f(n/gcd(n,k))^(mu(gcd(n,k))/phi(n/gcd(n,k))), where mu = A008683. (End)
a(n+1) is the number of binary words with exactly n distinct subsequences (when n > 0). - Radoslaw Zak, Nov 29 2021

			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
From _Eduard I. Vatutin_, Nov 01 2020: (Start)
The a(5)=4 cyclic Latin squares with the first row in ascending order are:
  0 1 2 3 4   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4
  1 2 3 4 0   2 3 4 0 1   3 4 0 1 2   4 0 1 2 3
  2 3 4 0 1   4 0 1 2 3   1 2 3 4 0   3 4 0 1 2
  3 4 0 1 2   1 2 3 4 0   4 0 1 2 3   2 3 4 0 1
  4 0 1 2 3   3 4 0 1 2   2 3 4 0 1   1 2 3 4 0
(End)

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.
Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 409.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 193.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 154-156.
C. W. Curtis, Pioneers of Representation Theory ..., Amer. Math. Soc., 1999; see p. 3.
J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Ellipses, Paris, 2004, Problème 529, pp. 71-257.
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.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B36.
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.
Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.2.21 pp. 281-294.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, New York, Heidelberg, Berlin, 2 vols., 1976, Vol. II, problem 71, p. 126.
Paulo Ribenboim, The New Book of Prime Number Records.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 28-33.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 162-167.

Daniel Forgues, Table of n, phi(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972.
Dario A. 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).
Alexander Bogomolny, Euler Function and Theorem.
Chris K. Caldwell, The Prime Glossary, Euler's phi function
Robert 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 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. [Annotated copy with A-numbers]
Kevin 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.
Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, arXiv:1201.3139 [math.NT]
M. Lal and P. Gillard, Table of Euler's phi function, n < 10^5, Math. Comp., 23 (1969), 682-683.
Derrick 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.
R. J. Mathar, Graphical representation among sequences closely related to this one (cf. N. J. A. Sloane, "Families of Essentially Identical Sequences").
Mathematics Stack Exchange, Is the Euler phi function bounded below? (2013).
Mathforum, Proving phi(m) Is Even.
Keith 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 Erdős 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 and 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.
Wacław F. Sierpiński, Euler's Totient Function And The Theorem Of Euler.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 14.
Ulrich Sondermann, Euler's Totient Function.
William A. Stein, Phi is a Multiplicative Function
Pinthira Tangsupphathawat, Takao Komatsu and Vichian Laohakosol, Minimal Polynomials of Algebraic Cosine Values, II, J. Int. Seq., Vol. 21 (2018), Article 18.9.5.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.
G. Villemin, Totient d'Euler.
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).
Gang Xiao, Numerical Calculator, To display phi(n) operate on "eulerphi(n)".
Index entries for "core" sequences
Index to divisibility sequences

Cf. A002088 (partial sums), A008683, A003434 (steps to reach 1), A007755, A049108, A002202 (values), A011755 (Sum k*phi(k)).
Cf. also 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.
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), A306411 (e=6), A239442 (e=7), A306412 (e=8), A239443 (e=9).
Cf. A003558, A135303.
Cf. A152455, A080737.
Cf. A076479.
Cf. A023900 (Dirichlet inverse of phi), A306633 (Dgf at s=3).

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

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

# Computes the first N terms of the sequence.
function A000010List(N)
    phi = [i for i in 1:N + 1]
    for i in 2:N + 1
        if phi[i] == i
            for j in i:i:N + 1
                phi[j] -= div(phi[j], i)
    end end end
return phi end
println(A000010List(68))  # Peter Luschny, Sep 03 2023

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

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
# Alternative without library function:
A000010List := proc(N) local i, j, phi;
    phi := Array([seq(i, i = 1 .. N+1)]);
    for i from 2 to N + 1 do
        if phi[i] = i then
            for j from i by i to N + 1 do
                phi[j] := phi[j] - iquo(phi[j], i) od
        fi od;
return phi end:
A000010List(68);  # Peter Luschny, Sep 03 2023

Mathematica
```
Array[EulerPhi, 70]
```

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

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

from sympy.ntheory import totient
print([totient(i) for i in range(1, 70)])  # Indranil Ghosh, Mar 17 2017

# Note also the implementation in A365339.

def A000010(n): return euler_phi(n) # Jaap Spies, Jan 07 2007

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

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: 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
a(n) = A096396(n) + A096397(n). - Reinhard Zumkeller, Mar 24 2012
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
From Bernard Schott, Nov 28 2020: (Start)
Sum_{m=1..n} 1/a(m) = A028415(n)/A048049(n) -> oo when n->oo.
Sum_{n >= 1} 1/a(n)^2 = A109695.
Sum_{n >= 1} 1/a(n)^3 = A335818.
Sum_{n >= 1} 1/a(n)^k is convergent iff k > 1.
a(2n) = a(n) iff n is odd, and, a(2n) > a(n) iff n is even. (End) [Actually, a(2n) = 2*a(n) for even n. - Jianing Song, Sep 18 2022]
a(n) = 2*A023896(n)/n, n > 1. - Richard R. Forberg, Feb 03 2021
From Richard L. Ollerton, May 09 2021: (Start)
For n > 1, Sum_{k=1..n} phi^{(-1)}(n/gcd(n,k))*a(gcd(n,k))/a(n/gcd(n,k)) = 0, where phi^{(-1)} = A023900.
For n > 1, Sum_{k=1..n} a(gcd(n,k))*mu(rad(gcd(n,k)))*rad(gcd(n,k))/gcd(n,k) = 0.
For n > 1, Sum_{k=1..n} a(gcd(n,k))*mu(rad(n/gcd(n,k)))*rad(n/gcd(n,k))*gcd(n,k) = 0.
Sum_{k=1..n} a(gcd(n,k))/a(n/gcd(n,k)) = n. (End)
a(n) = Sum_{d|n, e|n} gcd(d, e)*mobius(n/d)*mobius(n/e) (the sum is a multiplicative function of n by Tóth, and takes the value p^e - p^(e-1) for n = p^e, a prime power). - Peter Bala, Jan 22 2024
Sum_{n >= 1} phi(n)*x^n/(1 + x^n) = x + 3*x^3 + 5*x^5 + 7*x^7 + ... = Sum_{n >= 1} phi(2*n-1)*x^(2*n-1)/(1 - x^(4*n-2)). For the first equality see Pólya and Szegő, problem 71, p. 126. - Peter Bala, Feb 29 2024
Conjecture: a(n) = lim_{k->oo} (n^(k + 1))/A000203(n^k). - Velin Yanev, Dec 04 2024 [A000010(p) = p-1, A000203(p^k) = (p^(k+1)-1)/(p-1), so the conjecture is true if n is prime. - Vaclav Kotesovec, Dec 19 2024]

A006512 Greater of twin primes.

Original entry on oeis.org

5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609

Offset: 1

N. J. A. Sloane

Also primes that are the sum of two primes (which is possible only if 2 is one of the primes). - Cino Hilliard, Jul 02 2004, edited by M. F. Hasler, Nov 14 2019
The set of greater of twin primes larger than five is a proper subset of the set of primes of the form 3n + 1 (A002476). - Paul Muljadi, Jun 05 2008
Smallest prime > n-th isolated composite. - Juri-Stepan Gerasimov, Nov 07 2009
Subsequence of A175075. Union of a(n) and sequence A175080 is A175075. - Jaroslav Krizek, Jan 30 2010
A164292(a(n))=1; A010051(a(n)+2)=0 for n > 1. - Reinhard Zumkeller, Mar 29 2010
Omega(n) = Omega(n-2); d(n) = d(n-2). - Juri-Stepan Gerasimov, Sep 19 2010
Aside from the first term, all subsequent terms have digital root 1, 4, or 7. - J. W. Helkenberg, Jul 24 2013
Also primes p with property that the sum of the successive gaps between primes <= p is a prime number. - Robert G. Wilson v, Dec 19 2014
The phrase "x is an element of the {primes, positive integers} and there {exist no, exist} elements a,b of {1 and primes, primes}: a+b=x" determines A133410, A067829, A025584, A006512, A166081, A014092, A014091 and A038609 for the first few hundred terms with only de-duplication or omitting/including 3, 4 and 6 in the case of A166081/A014091 and one case of omitting/including 3 given 1 isn't prime. - Harry G. Coin, Nov 25 2015
The yet unproved Twin Prime Conjecture states that this sequence is infinite. - M. F. Hasler, Nov 14 2019

See A001359 for further references and links.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
Harvey Dubner, Twin Prime Statistics, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.2.
R. K. Guy, Letter to N. J. A. Sloane, Jun 1991
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 114.
Wikipedia, Twin prime.
Index entries for primes, gaps between

Subsequence of A139690.
Bisection of A077800.
Cf. A001097, A001359, A014574, A067829, A002476.

a006512 = (+ 2) . a001359 -- Reinhard Zumkeller, Feb 10 2015

[n: n in PrimesUpTo(1610)|IsPrime(n-2)]; // Bruno Berselli, Feb 28 2011

for i from 1 to 253 do if ithprime(i+1) = ithprime(i) + 2 then print({ithprime(i+1)}); fi; od; # Zerinvary Lajos, Mar 19 2007
P := select(isprime,[$1..1609]): select(p->member(p-2,P),P); # Peter Luschny, Mar 03 2011
A006512 := proc(n)
    2+A001359(n) ;
end proc: # R. J. Mathar, Nov 26 2014

Select[Prime[Range[254]], PrimeQ[# - 2] &] (* Robert G. Wilson v, Jun 09 2005 *)
Transpose[Select[Partition[Prime[Range[300]], 2, 1], Last[#] - First[#] == 2 &]][[2]] (* Harvey P. Dale, Nov 02 2011 *)
Cases[Prime[Range[500]] + 2, ?PrimeQ] (* _Fred Patrick Doty, Aug 23 2017 *)

PARI
```
select(p->isprime(p-2),primes(1000))
```

a(n)=p=3; while(p+2 < (p=nextprime(p+1)) || n-->0, ); p
vector(100, n, a(n)) \\ Altug Alkan, Dec 04 2015

from sympy import primerange, isprime
print([n for n in primerange(1, 2001) if isprime(n - 2)]) # Indranil Ghosh, Jul 20 2017

A057635 a(n) is the largest m such that phi(m) = n, where phi is Euler's totient function = A000010, or a(n) = 0 if no such m exists.

Original entry on oeis.org

2, 6, 0, 12, 0, 18, 0, 30, 0, 22, 0, 42, 0, 0, 0, 60, 0, 54, 0, 66, 0, 46, 0, 90, 0, 0, 0, 58, 0, 62, 0, 120, 0, 0, 0, 126, 0, 0, 0, 150, 0, 98, 0, 138, 0, 94, 0, 210, 0, 0, 0, 106, 0, 162, 0, 174, 0, 118, 0, 198, 0, 0, 0, 240, 0, 134, 0, 0, 0, 142, 0, 270, 0, 0, 0, 0, 0, 158, 0, 330, 0

Offset: 1

Jud McCranie, Oct 10 2000

nonn

To check that a property P holds for all EulerPhi(x) not exceeding n, for n with a(n) > 0, it suffices to check P for all EulerPhi(x) with x not exceeding a(n). - Joseph L. Pe, Jan 10 2002
The Alekseyev link in A131883 establishes the following explicit relationship between A131883, A036912 and A057635: for t belonging to A036912, we have t = A131883(A057635(t)-1). In other words, A036912(n) = A131883(A057635(A036912(n))-1) for all n.
From Jianing Song, Feb 16 2019: (Start)
Let f(n) = exp(gamma)*log(log(n)) + 2.5/log(log(n)), then a(n) < n*f(n^2) for all n > 1, where gamma = A001620.
Proof. Without loss of generality we suppose log(log(n)) > n_0 = sqrt(2.5/exp(gamma)) = 1.18475..., then f(n), n/f(n) and N(n) = ceiling(n*f(n^2)) are all monotonically increasing functions of n, and we have f(n) < 2*exp(gamma)*log(log(n)).
By the formula (3.41) in Theorem 15 by J. Barkley Rosser and Lowell Schoenfeld we have phi(k) > k/f(k) for k != 1, 2, 223092870. N(31802157) = 223092869 < 223092870, N(31802158) = 223092877 > 223092870, so N(n) != 223092870 (N(n) is increasing). So phi(N(n)) > N(n)/f(N(n)) > (n*f(n^2))/f(n*f(n^2)) (n/f(n) is increasing and log(log(n*f(n^2))) > n_0).
Note that f(n^2) < 2*exp(gamma)*log(log(n^2)) < 2*exp(gamma)*(log(n^2)/e) = 4*exp(gamma-1)*log(n) < 4*exp(gamma-2)*n < n, so n*f(n^2) < n^2, f(n*f(n^2)) < f(n^2) (f(n) is increasing and log(log(n*f(n^2))) > n_0), so phi(N(n)) > n. As a result, a(n) <= N(n) - 1 < n*f(n^2).
Conjecturally a(n) < n*f(n) for all n > 2. (End)

			m = 12 is the largest value of m such that phi(m) = 4, so a(4) = 12.

T. D. Noe, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1) (1962), 64-94.

Cf. A000010, A005277, A014197, A049283.

Cf. A006511 (largest k for which A000010(k) = A002202(n)).

a = Table[0, {100}]; Do[ t = EulerPhi[n]; If[t < 101, a[[t]] = n], {n, 1, 10^6}]; a

a(n) = if(n%2, 2*(n==1), forstep(k=floor(exp(Euler)*n*log(log(n^2))+2.5*n/log(log(n^2))), n, -1, if(eulerphi(k)==n, return(k)); if(k==n, return(0)))) \\ Jianing Song, Feb 15 2019

apply( {A057635(n,m=istotient(n))=if(!m, 0, n>1, m=log(log(n)*2); m=bitand(n*(exp(Euler)*m+2.5/m)\1,-2); while(eulerphi(m)!=n, m-=2); m, 2)}, [1..99]) \\ If n is known to be a totient, a nonzero 2nd arg can be given to avoid the check. - M. F. Hasler, Aug 13 2021

a(n) = invphiMax(n); \\ Amiram Eldar, Nov 14 2024 using Max Alekseyev's invphi.gp

a(2n+1) = 0 for n > 0, and a(2n) = 0 iff 2n is in A005277.

Edited and escape clause added to definition by M. F. Hasler, Aug 13 2021

A058277 Number of values of k such that phi(k) = n, where n runs through the values (A002202) taken by phi.

Original entry on oeis.org

2, 3, 4, 4, 5, 2, 6, 6, 4, 5, 2, 10, 2, 2, 7, 8, 9, 4, 3, 2, 11, 2, 2, 3, 2, 9, 8, 2, 2, 17, 2, 10, 2, 6, 6, 3, 17, 4, 2, 3, 2, 9, 2, 6, 3, 17, 2, 9, 2, 7, 2, 2, 3, 21, 2, 2, 7, 12, 4, 3, 2, 12, 2, 8, 2, 10, 4, 2, 21, 2, 2, 8, 3, 4, 2, 3, 19, 5, 2, 8, 2, 2, 6, 2, 31, 2, 9, 10

Offset: 1

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 05 2001

Carmichael (1922) conjectured that the number 1 never appears in this sequence. Sierpiński conjectured and Ford (1998) proved that all integers greater than 1 occur in the sequence. Erdős (1958) proved that if s >= 1 appears in the sequence then it appears infinitely often. - Nick Hobson, Nov 04 2006

A002202(n) occurs a(n) times in A007614. - Reinhard Zumkeller, Nov 22 2015

Édouard Lucas, Théorie des Nombres, Blanchard 1958.

T. D. Noe, Table of n, a(n) for n=1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
R. D. Carmichael, Note on Euler's totient function, Bull. Amer. Math. Soc. 28 (1922), pp. 109-110.
Paul Erdős, Some remarks on Euler's totient function, Acta Arith. 4 (1958), pp. 10-19.
M. Farrokhi D. G., Gap function to compute the inverse of Euler's totient function.
Kevin Ford, The distribution of totients, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 27-34.
Nick Hobson, Solution to puzzle 152: Totient valence.
Eric Weisstein's World of Mathematics, Totient Valence Function.

The nonzero terms of A014197.
Cf. A000010, A002202, A007614.
Cf. A006511 (largest k for which A000010(k) = A002202(n)).

import Data.List (group)
a058277 n = a058277_list !! (n-1)
a058277_list = map length $ group a007614_list
-- Reinhard Zumkeller, Nov 22 2015

max = 300; inversePhi[?OddQ] = {}; inversePhi[1] = {1, 2}; inversePhi[m] := Module[{p, nmax, n, nn}, p = Select[Divisors[m] + 1, PrimeQ]; nmax = m * Times @@ (p/(p-1)); n = m; nn = Reap[While[n <= nmax, If[EulerPhi[n] == m, Sow[n]]; n++]] // Last; If[nn == {}, {}, First[nn] ] ]; Reap[For[n = 1, n <= max, n = If[n == 1, 2, n+2], nn = inversePhi[n] ; If[nn != {} , Sow[nn // Length] ] ] ] // Last // First (* Jean-François Alcover, Nov 21 2013 *)

lista(nmax) = {my(m); for(n = 1, nmax, m = invphiNum(n); if(m > 0, print1(m, ", ")));} \\ Amiram Eldar, Nov 15 2024 using Max Alekseyev's invphi.gp

More terms from Nick Hobson, Nov 04 2006

A002181 Least number k such that phi(k) = m, where m runs through the values (A002202) taken by phi.

Original entry on oeis.org

1, 3, 5, 7, 15, 11, 13, 17, 19, 25, 23, 35, 29, 31, 51, 37, 41, 43, 69, 47, 65, 53, 81, 87, 59, 61, 85, 67, 71, 73, 79, 123, 83, 129, 89, 141, 97, 101, 103, 159, 107, 109, 121, 113, 177, 143, 127, 255, 131, 161, 137, 139, 213, 185, 149, 151, 157, 187, 163, 249, 167, 203, 173

Offset: 1

N. J. A. Sloane

nonn

Inverse of Euler totient function.
A051445 without the zeros. The values of m are in A002180.
According to Guy, the first even term is for 2m = 16842752 = 257*2^16. If there are only five Fermat primes, then terms will be even for 2m = 2^r for all r > 31. This was discussed in problem E3361. - T. D. Noe, Aug 14 2008

J. W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 64.
R. K. Guy, Unsolved problems in number theory, B39.
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).

T. D. Noe, Table of n, a(n) for n = 1..10000
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]
T. D. Noe, Numbers Like 16842752.
William P. Wardlaw, L. L. Foster and R. J. Simpson, Problem E3361, Amer. Math. Monthly, Vol. 98, No. 5 (May, 1991), 443-444.
K. W. Wegner, Values of phi(x) = n for n from 2 through 1978, mimeographed manuscript, no date [Annotated scanned copy]

Cf. A058277, A006511, A002202, A061026.

With[{ep=EulerPhi[Range[1000]]},Flatten[Table[Position[ep,n,{1},1],{n,200}]]] (* Harvey P. Dale, Apr 10 2015 *)

a(n) = A061026(A002202(n)). - Flávio V. Fernandes, Oct 08 2023

Offset and initial term corrected Oct 07 2007

Revised definition from T. D. Noe, Aug 14 2008

A036913 Sparsely totient numbers; numbers n such that m > n implies phi(m) > phi(n).

Original entry on oeis.org

2, 6, 12, 18, 30, 42, 60, 66, 90, 120, 126, 150, 210, 240, 270, 330, 420, 462, 510, 630, 660, 690, 840, 870, 1050, 1260, 1320, 1470, 1680, 1890, 2310, 2730, 2940, 3150, 3570, 3990, 4620, 4830, 5460, 5610, 5670, 6090, 6930, 7140, 7350, 8190, 9240, 9660

Offset: 1

David W. Wilson

nonn

The paper by Masser and Shiu lists 150 terms of this sequence less than 10^6. For odd prime p, they show that p# and p*p# are in this sequence, where p# denotes the primorial (A002110). - T. D. Noe, Jun 14 2006

Conjecture: Except for 2 and 18, all terms are Zumkeller numbers (A083207). Verified for the first 1800 terms. - Ivan N. Ianakiev, Sep 04 2022

			This sequence contains 60 because of all the numbers whose totient is <=16, 60 is the largest such number. [From _Graeme McRae_, Feb 12 2009]
From _Michael De Vlieger_, Jun 25 2017: (Start)
Positions of primorials A002110(k) in a(n):
     n     k       a(n) = A002110(k)
  ----------------------------------
     1     1                       2
     2     2                       6
     5     3                      30
    13     4                     210
    31     5                    2310
    69     6                   30030
   136     7                  510510
   231     8                 9699690
   374     9               223092870
   578    10              6469693230
   836    11            200560490130
  1169    12           7420738134810
  1591    13         304250263527210
  2149    14       13082761331670030
  2831    15      614889782588491410
  3667    16    32589158477190044730
  4661    17  1922760350154212639070
(End)

T. D. Noe, Table of n, a(n) for n = 1..5000
Roger C. Baker and Glyn Harman, Sparsely totient numbers, Annales de la Faculté des Sciences de Toulouse Ser. 6, 5 no. 2 (1996), 183-190.
Glyn Harman, On sparsely totient numbers, Glasgow Math. J. 33 (1991), 349-358.
D. W. Masser and P. Shiu, On sparsely totient numbers, Pacific J. Math. 121, no. 2 (1986), 407-426.
Michael De Vlieger, Largest k such that A002110(k) | a(n) and A287352(a(n)).
Michael De Vlieger, First term m > prime(n)^2 in A036913 such that gcd(prime(n), m) = 1.

Cf. A097942 (highly totient numbers). Records in A006511 (see also A132154).

nn=10000; lastN=Table[0,{nn}]; Do[e=EulerPhi[n]; If[e<=nn, lastN[[e]]=n], {n,10nn}]; mx=0; lst={}; Do[If[lastN[[i]]>mx, mx=lastN[[i]]; AppendTo[lst,mx]], {i,Length[lastN]}]; lst (* T. D. Noe, Jun 14 2006 *)

A032446 Number of solutions to phi(k) = 2n.

Original entry on oeis.org

3, 4, 4, 5, 2, 6, 0, 6, 4, 5, 2, 10, 0, 2, 2, 7, 0, 8, 0, 9, 4, 3, 2, 11, 0, 2, 2, 3, 2, 9, 0, 8, 2, 0, 2, 17, 0, 0, 2, 10, 2, 6, 0, 6, 0, 3, 0, 17, 0, 4, 2, 3, 2, 9, 2, 6, 0, 3, 0, 17, 0, 0, 2, 9, 2, 7, 0, 2, 2, 3, 0, 21, 0, 2, 2, 0, 0, 7, 0, 12, 4, 3, 2, 12, 0, 2, 0, 8, 2, 10

Offset: 1

Ursula Gagelmann (gagelmann(AT)altavista.net)

By Carmichael's conjecture, a(n) <> 1 for any n. See A074987. - Thomas Ordowski, Sep 13 2017

a(n) = 0 iff n is a term of A079695. - Bernard Schott, Oct 02 2021

			If n = 8 then phi(x) = 2*8 = 16 is satisfied for only a(8) = 6 values of x, viz. 17, 32, 34, 40, 48, 60.

Albert H. Beiler, Recreations in the Theory of Numbers, The Queen of Mathematics Entertains, Second Edition, Dover Publications, Inc., NY, 1966, page 90.

T. D. Noe, Table of n, a(n) for n = 1..5000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Matteo Caorsi and Sergio Cecotti, Geometric classification of 4d N=2 SCFTs, arXiv:1801.04542 [hep-th], 2018.
Carl Pomerance, Popular values of Euler's function, Mathematica 27 (1980), 84-89.

Bisection of A014197.
Cf. A000010, A005277, A074987, A079695, A085758.
Cf. A006511 (largest k for which A000010(k) = A002202(n)), A057635.

[#EulerPhiInverse( 2*n):n in [1..100]]; // Marius A. Burtea, Sep 08 2019

with(numtheory); [ seq(nops(invphi(2*n)), n=1..90) ];

t = Table[0, {100} ]; Do[a = EulerPhi[n]; If[a < 202, t[[a/2]]++ ], {n, 3, 10^5} ]; t

a(n) = invphiNum(2*n); \\ Amiram Eldar, Nov 15 2024 using Max Alekseyev's invphi.gp

Extended by Robin Trew (trew(AT)hcs.harvard.edu).

A036912 Indices of the left-to-right maxima in A057635.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 20, 24, 32, 36, 40, 48, 64, 72, 80, 96, 120, 128, 144, 160, 176, 192, 224, 240, 288, 320, 336, 384, 432, 480, 576, 672, 720, 768, 864, 960, 1056, 1152, 1280, 1296, 1344, 1440, 1536, 1680, 1728, 1920, 2112, 2208, 2304, 2400, 2592, 2688

Offset: 1

David W. Wilson

nonn

A number m belongs to this sequence iff A057635(k) < A057635(m) for all k

Indices of records in A057635(n), the maximal m with phi(m)=n.

The Alekseyev link in A131883 establishes the following explicit relationship between A131883, A036912 and A057635. Namely, for t belonging to A036912, we have t=A131883(A057635(t)-1). In other words, A036912(n) = A131883(A057635(A036912(n))-1) for all n.

T. D. Noe, Table of n, a(n) for n = 1..5000

Cf. A057635, A000010, A006511, A036913, A132154, A131883.

Block[{nn = 10^6, s, t, u}, s = PositionIndex@ Array[EulerPhi, nn]; t = ConstantArray[0, nn]; u = Take[ReplacePart[t, Map[# -> Last@ Lookup[s, #] &, Keys@ s]], 10^(Log10[nn] - 2)]; Map[FirstPosition[u, #][[1]] &, Union@ FoldList[Max, u]]] (* Michael De Vlieger, Oct 24 2017 *)

a(n) = A000010(A036913(n)). - Max Alekseyev, Nov 07 2007

More precise definition from Max Alekseyev, Nov 07 2007

A206864 Prime numbers of the form Phi_k(m), where k > 2, |m| > 1, and Phi_k(m) is the k-th cyclotomic polynomial evaluated at m.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 31, 37, 43, 61, 73, 101, 127, 151, 157, 197, 211, 241, 257, 307, 331, 401, 421, 463, 521, 547, 577, 601, 677, 683, 757, 1093, 1123, 1297, 1483, 1601, 1723, 2551, 2731, 2801, 2917, 2971, 3137, 3307, 3541, 3907, 4357, 4423, 4561, 4831, 5113

Offset: 1

Lei Zhou, Feb 13 2012

nonn

These are the prime numbers picked from sequence A206942.
Choosing negative m does not generate more primes, so it does not need negative m part in the Mathematica program.
The provided mathematica program generate this sequence in six steps:
Step 1: Find the minimum m such that Phi(6, m) is greater than the search boundary maxdata, and adjust the search boundary to the next: ( maxdata = 5200; max = Ceiling[(1 + Sqrt[1 + 4*(maxdata - 1)])/2]; )
Step 2: Find the even number eulerbound such that 2^(eulerbound+1)-1 > maxdata: ( eulerbound = 2*Floor[(Log[2, maxdata])/2 + 0.5]; )
  This is the maximum possible value of Phi(k, 2) when Phi(k, m) has a totient function value of eulerbound;
Step 3: Adjust (up) the eulerbound such that it is an element of A002202 and find the group of ks such that Phi(k, m) has the same totient function value eulerbound: ( phiinv[n_, pl_] := Module[{i, p, e, pe, val}, If[pl == {}, Return[If[n == 1, {1}, {}]]]; val = {}; p = Last[pl]; For[e = 0; pe = 1, e == 0 || Mod[n, (p - 1) pe/p] == 0, e++; pe *= p, val = Join[val, pe*phiinv[If[e == 0, n, n*p/pe/(p - 1)], Drop[pl, -1]]]]; Sort[val]]; phiinv[n_] := phiinv[n, Select[1 + Divisors[n], PrimeQ]]; While[eulergroup = phiinv[eulerbound]; lu = Length[eulergroup]; lu == 0, eulerbound = eulerbound + 2]; )
Step 4: Make list of k values such that the totient function of Phi(k, m) smaller or equal to the chosen Euler boundary eulerbound, and sort it in the order of the Phi(k, 2): ( Select[Range[eulergroup[[Length[eulergroup]]]], EulerPhi[#] <= eulerbound &]; ap = SortBy[t, Cyclotomic[#, 2] &])
Step 5: Scan k in the set of ap, 1 < m <= max, for all appeared primes that are smaller than maxdata: (a = {}; Do[i = 2; While[i++; cc = Cyclotomic[ap[[i]], m]; cc <= maxdata, If[PrimeQ[cc], a = Append[a, cc]]], {m, 2, max}];)
Step 6: Remove duplicate and sort the set generated in the above step: ( Sort[DeleteDuplicates[a]] )
Through these steps, a mathematically abundant algorithm is presented to find all the terms up to an arbitrary bound, without requiring the user to determine any other search parameters.

			Prime 3 = Phi_6(2); so a(1) = 3;
Prime 5 = Phi_4(2), so a(2) = 5;
...
Prime 17 = Phi_8(2), so a(6)=17;
Primes 19 and 23 are not in A206942;
Prime 31 = Phi_5(2), so a(7)=31.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Cf. A206942, A006511.

# Function isA206942 is defined in A206942.
L = [n for n in 1:5113 if isprime(ZZ(n)) && isA206942(n)]
println(L) # Peter Luschny, Feb 21 2018

maxdata = 5200; max = Ceiling[(1 + Sqrt[1 + 4*(maxdata - 1)])/2]; eulerbound = 2*Floor[(Log[2, maxdata])/2 + 0.5]; phiinv[n_, pl_] := Module[{i, p, e, pe, val}, If[pl == {}, Return[If[n == 1, {1}, {}]]]; val = {}; p = Last[pl]; For[e = 0; pe = 1, e == 0 || Mod[n, (p - 1) pe/p] == 0, e++; pe *= p, val = Join[val, pe*phiinv[If[e == 0, n, n*p/pe/(p - 1)], Drop[pl, -1]]]]; Sort[val]]; phiinv[n_] := phiinv[n, Select[1 + Divisors[n], PrimeQ]]; While[eulergroup = phiinv[eulerbound]; lu = Length[eulergroup]; lu == 0, eulerbound = eulerbound + 2]; t = Select[Range[eulergroup[[Length[eulergroup]]]], EulerPhi[#] <= eulerbound &]; ap = SortBy[t, Cyclotomic[#, 2] &]; a = {}; Do[i = 2; While[i++; cc = Cyclotomic[ap[[i]], m]; cc <= maxdata, If[PrimeQ[cc], a = Append[a, cc]]], {m, 2, max}]; Sort[DeleteDuplicates[a]]
(* Alternatively: *)
isA206864[n_] := If[! PrimeQ[n], Return[False],
    K = Floor[5.383 Log[n]^1.161]; M = Floor[2 Sqrt[n/3]];
    For[k = 3, k <= K, k++, For[x = 2, x <= M, x++,
        If[n == Cyclotomic[k, x], Return[True]]]];
    Return[False]
]; Select[Range[1000], isA206864] (* Peter Luschny, Feb 21 2018 *)

Showing 1-10 of 19 results. Next

cp's OEIS Frontend

A132154 Where records occur in A006511.

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A000010 Euler totient function phi(n): count numbers <= n and prime to n.

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A006512 Greater of twin primes.

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A057635 a(n) is the largest m such that phi(m) = n, where phi is Euler's totient function = A000010, or a(n) = 0 if no such m exists.

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A058277 Number of values of k such that phi(k) = n, where n runs through the values (A002202) taken by phi.

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A002181 Least number k such that phi(k) = m, where m runs through the values (A002202) taken by phi.

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