A002618 a(n) = n*phi(n).

1, 2, 6, 8, 20, 12, 42, 32, 54, 40, 110, 48, 156, 84, 120, 128, 272, 108, 342, 160, 252, 220, 506, 192, 500, 312, 486, 336, 812, 240, 930, 512, 660, 544, 840, 432, 1332, 684, 936, 640, 1640, 504, 1806, 880, 1080, 1012, 2162, 768, 2058, 1000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• mult
• look

Comments

Also Euler phi function of n^2.

For n >= 3, a(n) is also the size of the automorphism group of the dihedral group of order 2n. This automorphism group is isomorphic to the group of transformations x -> ax + b, where a, b and x are integers modulo n and a is coprime to n. Its order is n*phi(n). - Ola Veshta (olaveshta(AT)my-deja.com), Mar 18 2001

Order of metacyclic group of polynomial of degree n. - Artur Jasinski, Jan 22 2008

It appears that this sequence gives the number of permutations of 1, 2, 3, ..., n that are arithmetic progressions modulo n. - John W. Layman, Aug 27 2008

The conjecture by Layman is correct. Obviously any such permutation must have an increment that is prime to n, and almost as obvious that any such increment will work, with any starting value; hence phi(n) * n total. - Franklin T. Adams-Watters, Jun 09 2009

Consider the numbers from 1 to n^2 written line by line as an n X n square: a(n) = number of numbers that are coprime to all their horizontal and vertical immediate neighbors. - Reinhard Zumkeller, Apr 12 2011

n -> a(n) is injective: a(m) = a(n) implies m = n. - Franz Vrabec, Dec 12 2012 (See Mathematics Stack Exchange link for a proof.)

a(p) = p*(p-1) a pronic number, see A036689 and A002378. - Fred Daniel Kline, Mar 30 2015

Conjecture: All the rational numbers Sum_{i=j..k} 1/a(i) with 0 < min{2,k} <= j <= k have pairwise distinct fractional parts. - Zhi-Wei Sun, Sep 24 2015

From Jianing Song, Aug 25 2023: (Start)

a(n) is the order of the holomorph (see the Wikipedia link) of the cyclic group of order n. Note that Hol(C_n) and Aut(D_{2n}) are isomorphic unless n = 2, where D_{2n} is the dihedral group of order 2*n. See the Wordpress link.

The odd-indexed terms form a subsequence of A341298: the holomorph of an abelian group of odd order is a complete group. See Theorem 3.2, Page 618 of the W. Peremans link. (End)

Examples

			a(4) = 8 since phi(4) = 2 and 4 * 2 = 8.
a(5) = 20 since phi(5) = 4 and 5 * 4 = 20.
		

References

• Peter Giblin, Primes and Programming: An Introduction to Number Theory with Computing. Cambridge: Cambridge University Press (1993) p. 116, Exercise 1.10.
• J. L. Lagrange, Oeuvres, Vol. III Paris 1869.
• 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).

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Daniel Fischer, answer to Injectivity of the function n times the Euler Totient of n, Mathematics Stack Exchange, October 2013.
• Mikhail R. Gabdullin and Vitalii V. Iudelevich, Numbers of the form kf(k), arXiv:2201.09287 [math.NT] (2022).
• Dmitry Krachun and Zhi-Wei Sun, Each positive rational number has the form phi(m^2)/phi(n^2), arXiv:2001.03736 [math.HO], 2020. See also The American Mathematical Monthly (2020) Vol. 127, Issue 9, 847-849.
• F. Luca and A. O. Munagi, The number of permutations which form arithmetic progressions modulo m, Annals of the Alexandru Ioan Cuza University, 2014, DOI: 10.2478/aicu-2014-0053. [Broken link]
• C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
• W. Peremans, Completeness of Holomorphs, Nederl. Akad. Wetensch. Indag. Math. Proc. Ser. A, 60. (1957) 608-619.
• N. J. A. Sloane, Notes on A002618, A002619, etc.
• J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
• J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61. [Annotated scanned copy. Note that the scanned pages are out of order]
• Wikipedia, Holomorph.
• Wordpress, Automorphisms of dihedral groups.

Crossrefs

First column of A047916.

Cf. A002619, A011755 (partial sums), A047918, A000010, A053650, A053191, A053192, A036689, A058161, A009262, A082473 (same terms, sorted into ascending order), A256545, A327172 (a left inverse), A327173, A065484.

Subsequence of A323333.

Programs

Haskell

a002618 n = a000010 n * n  -- Reinhard Zumkeller, Dec 21 2012

Magma

[n*EulerPhi(n): n in [1..150]]; // Vincenzo Librandi, Apr 04 2011

Maple

with(numtheory):a:=n->phi(n^2): seq(a(n), n=1..50); # Zerinvary Lajos, Oct 07 2007

Mathematica

Table[n EulerPhi[n], {n, 100}] (* Artur Jasinski, Jan 22 2008 *)

MuPAD

numlib::phi(n^2)$ n=1..81 // Zerinvary Lajos, May 13 2008

PARI

a(n)=n*eulerphi(n) \\ Charles R Greathouse IV, Nov 20 2012

Python

from sympy import totient as phi
def a(n): return n*phi(n)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Mar 16 2022

Sage

[euler_phi(n^2) for n in range(1,51)] # Zerinvary Lajos, Jun 06 2009

Formula

Multiplicative with a(p^e) = (p-1)*p^(2e-1). - David W. Wilson, Aug 01 2001

Dirichlet g.f.: zeta(s-2)/zeta(s-1). - R. J. Mathar, Feb 09 2011

a(n) = A173557(n) * A102631(n). - R. J. Mathar, Mar 30 2011

From Wolfdieter Lang, May 12 2011: (Start)

a(n)/2 = A023896(n), n >= 2.

a(n)/2 = (1/n) * Sum_{k=1..n-1, gcd(k,n)=1} k, n >= 2 (see A023896 and A076512/A109395). (End)

a(n) = lcm(phi(n^2),n). - Enrique Pérez Herrero, May 11 2012

a(n) = phi(n^2). - Wesley Ivan Hurt, Jun 16 2013

a(n) = A009195(n) * A009262(n). - Michel Marcus, Oct 24 2013

G.f.: -x + 2*Sum_{k>=1} mu(k)*k*x^k/(1 - x^k)^3. - Ilya Gutkovskiy, Jan 03 2017

a(n) = A082473(A327173(n)), A327172(a(n)) = n. -- Antti Karttunen, Sep 29 2019

Sum_{n>=1} 1/a(n) = 2.203856... (A065484). - Amiram Eldar, Sep 30 2019

Define f(x) = #{n <= x: a(n) <= x}. Gabdullin & Iudelevich show that f(x) ~ c*sqrt(x) for c = Product_{p prime} (1 + 1/(p*(p - 1 + sqrt(p^2 - p)))) = 1.3651304521525857... - Charles R Greathouse IV, Mar 16 2022

a(n) = Sum_{d divides n} A001157(d)*A046692(n/d); that is, the Dirichlet convolution of sigma_2(n) and the Dirichlet inverse of sigma_1(n). - Peter Bala, Jan 26 2024

Extensions

Better description from Labos Elemer, Feb 18 2000