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A002618 a(n) = n*phi(n).
(Formerly M1568 N0611)
75
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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

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 (first 1000 terms from T. D. Noe), Table of n, a(n) for n = 1..10000

F. Luca, 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.

C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.

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]

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 = (sum(k, k = 1..n - 1, with gcd(k, n) = 1))/n, 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

EXAMPLE

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

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

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 *)

PROG

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

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

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

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

(Haskell)

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

CROSSREFS

First column of A047916.

Cf. A002619, A047918, A000010, A053650, A053191, A053192, A036689, A058161, A009262, A256545.

Sequence in context: A183171 A124827 A140965 * A135616 A069553 A275826

Adjacent sequences:  A002615 A002616 A002617 * A002619 A002620 A002621

KEYWORD

nonn,easy,nice,mult,look

AUTHOR

N. J. A. Sloane

EXTENSIONS

Better description from Labos Elemer, Feb 18 2000

STATUS

approved

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Last modified October 20 07:17 EDT 2018. Contains 316378 sequences. (Running on oeis4.)