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A002618
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a(n) = n*phi(n).
(Formerly M1568 N0611)
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99
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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)
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OFFSET
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1,2
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COMMENTS
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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 StackExchange-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
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REFERENCES
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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).
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LINKS
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Michael De Vlieger (first 1000 terms from T. D. Noe), Table of n, a(n) for n = 1..10000
Daniel Fischer, answer to Injectivity of the function n times the Euler Totient of n, Math StackExchange, October 2013.
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, 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]
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FORMULA
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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
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
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EXAMPLE
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a(4) = 8 since phi(4) = 2 and 4 * 2 = 8.
a(5) = 20 since phi(5) = 4 and 5 * 4 = 20.
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MAPLE
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with(numtheory):a:=n->phi(n^2): seq(a(n), n=1..50); # Zerinvary Lajos, Oct 07 2007
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MATHEMATICA
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Table[n EulerPhi[n], {n, 100}] (* Artur Jasinski, Jan 22 2008 *)
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PROG
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(MuPAD) numlib::phi(n^2)$ n=1..81 // Zerinvary Lajos, May 13 2008
(Sage) [euler_phi(n^2) for n in range(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
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CROSSREFS
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First column of A047916.
Cf. A002619, 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.
Sequence in context: A183171 A124827 A140965 * A135616 A069553 A275826
Adjacent sequences: A002615 A002616 A002617 * A002619 A002620 A002621
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KEYWORD
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nonn,easy,nice,mult,look
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Better description from Labos Elemer, Feb 18 2000
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STATUS
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approved
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