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Search: id:A003277
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| A003277 |
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Cyclic numbers: n such that n and phi(n) are relatively prime; also n such that there is just one group of order n, i.e. A000001(n) = 1.
(Formerly M0650)
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+0
25
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| 1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, 145, 149, 151, 157, 159, 161, 163, 167, 173
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Except for a(2)=2, all the terms in the sequence are odd. This is because of the existence of a non-cyclic dihedral group of order 2n for each n>1. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 09 2001
Also GCD[n, A051593[n]] = 1 (Labos E.).
n such that x^n==1 (mod n) has no solution 2<=x<=n - Benoit Cloitre (benoit7848c(AT)orange.fr), May 10 2002
There is only one group (the cyclic group of order n) whose order is n. - Gerard P. Michon (g.michon(AT)att.net), Jan 08 2008
Any divisor of a Carmichael number (A002997) must be odd and cyclic. Conversely, G. P Michon conjectured (c. 1980) that any odd cyclic number has at least one Carmichael multiple (if the conjecture is true, each of them has infinitely many such multiples). In 2007, Michon & Crump produced explicit Carmichael multiples of all odd cyclic numbers below 10000 (see link). - Gerard P. Michon (g.michon(AT)att.net), Jan 08 2008
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
P. Erdos, Some asymptotic formulas in number theory. J. Indian Math. Soc. (N.S.) 12, (1948). 75-78.
J. Pakianathan and K. Shankar, Nilpotent numbers, Amer. Math. Monthly, 107 (Aug. 2000), 631-634.
J. S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 7.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Max Alekseyev, Michon's conjecture (Open Problem Garden, Aug. 2007).
Gerard P. Michon, Carmichael Divisors
G. P. Michon and J. K. Crump, Carmichael Multiples of Odd Cyclic Numbers (up to 10000)
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FORMULA
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n = p_1*p_2*...*p_k (for some k >= 0), where the p_i are distinct primes and no p_j-1 is divisible by any p_i.
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CROSSREFS
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Cf. A000010, A009195, A050384 (the same sequence but with the primes removed). Also A000001(n) = 1.
Cf. A002997 A054395.
Cf. A000001, A003277, A054395, A054396, A054397, A135850.
Adjacent sequences: A003274 A003275 A003276 this_sequence A003278 A003279 A003280
Sequence in context: A090421 A102553 A069161 this_sequence A117287 A121615 A097605
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com) and R. P. Stanley
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EXTENSIONS
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More terms from Christian G. Bower (bowerc(AT)usa.net).
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