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Search: id:A023153
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| A023153 |
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Number of cycles of function f(x) = x^2 mod n. |
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+0
9
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| 1, 2, 2, 2, 2, 4, 3, 2, 3, 4, 3, 4, 3, 6, 4, 2, 2, 6, 4, 4, 6, 6, 3, 4, 3, 6, 4, 6, 4, 8, 6, 2, 6, 4, 6, 6, 4, 8, 6, 4, 3, 12, 7, 6, 6, 6, 4, 4, 7, 6, 4, 6, 3, 8, 6, 6, 8, 8, 3, 8, 6, 12, 10, 2, 6, 12, 6, 4, 6, 12, 7, 6, 4, 8, 6, 8, 10, 12, 6, 4, 5, 6, 4, 12, 4, 14, 8, 6, 3, 12, 10, 6, 12, 8, 8, 4, 3, 14, 10, 6
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Not multiplicative; the smallest counterexample is a(63). - T. D. Noe (noe(AT)sspectra.com), Nov 14 2006
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REFERENCES
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Earle Blanton, Spencer Hurd and Judson McCranie, On the Digraph Defined by Squaring Mod m, When m Has Primitive Roots, Congressus Numerantium, vol. 82, 167-177, 1992.
Earle Blanton, Spencer Hurd and Judson McCranie, On the Digraph Defined by Squaring Mod n [prime n], Fib. Quarterly, vol. 30, #4, 1992, 322-334.
J. J. Brennan and B. Geist, Analysis of Iterated Modular Exponentiation: The Orbits of x alpha mod N, Designs, Codes and Cryptography 13, 229-245 (1998) (specially Th. 6 and 7).
G. Chasse, Applications d'un corps fini dans lui-meme, Univ. Rennes, Rennes, 1984; Math. Rev. 86e:11118.
T. D. Rogers, The graph of the square mapping on the prime fields, Discrete Math. 148 (1996), 317-324.
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LINKS
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David W. Wilson, Table of n, a(n) for n=1..10000
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FORMULA
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In case (Z/nZ)^* is cyclic there is a formula (see Chasse and Rogers). Let C_m denote the cyclic group of order m. Let a(m) denote the number of cycles in the graph of C_m relative to the mapping f. Then the number of cycles equals a(m)=\sum_{ d divides n }\varphi(d)/ord_d(2). - Pieter Moree, Jul 04, 2002
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CROSSREFS
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Cf. A023154-A023161 (cycles of the functions f(x)=x^k mod n for k=3..10)
Sequence in context: A049047 A037088 A064486 this_sequence A023159 A098983 A097576
Adjacent sequences: A023150 A023151 A023152 this_sequence A023154 A023155 A023156
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KEYWORD
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nonn
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AUTHOR
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David W. Wilson (davidwwilson(AT)comcast.net)
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