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A002618
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n*phi(n) (cf. A000010).
(Formerly M1568 N0611)
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37
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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; internal format)
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OFFSET
| 1,2
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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 (grafix(AT)csl.pl), Jan 22 2008
It appears that this sequence gives the number of permutations of 1,2,3...,n that are arithmetic progressions modulo n. [From John W. Layman (layman(AT)math.vt.edu), 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. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 09 2009]
Consider the numbers from 1 to n^2 written line by line as a 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]
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REFERENCES
| 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).
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
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FORMULA
| Multiplicative with a(p^e) = (p-1)*p^(2e-1). - David W. Wilson (davidwwilson(AT)comcast.net), 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)
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MAPLE
| with(numtheory):a:=n->phi(n^2): seq(a(n), n=1..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 07 2007
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MATHEMATICA
| Table[n EulerPhi[n], {n, 1, 100}] - Artur Jasinski (grafix(AT)csl.pl), Jan 22 2008
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PROG
| (MuPad) numlib::phi(n^2)$ n=1..81 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 13 2008
(Sage) [euler_phi(n^2) for n in xrange(1, 51)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 06 2009]
(MAGMA) [n*EulerPhi(n): n in [1..150]]; - Vincenzo Librandi, Apr 04 2011
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CROSSREFS
| First column of A047916. Cf. A002619, A047918.
Cf. A000010, A053650, A053191, A053192, A036689.
Cf. A058161.
Sequence in context: A183171 A124827 A140965 * A069553 A143481 A093968
Adjacent sequences: A002615 A002616 A002617 * A002619 A002620 A002621
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KEYWORD
| nonn,easy,nice,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Better description from Labos E. (labos(AT)ana.sote.hu ), Feb 18 2000.
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