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A005848
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Cyclotomic fields with class number 1 (or with unique factorization).
(Formerly M2304)
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3
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1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 48, 60, 84
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OFFSET
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1,2
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COMMENTS
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Note that if n == 2 (mod 4) Q(zeta_n) is the same field as Q(zeta_{n/2}), so this sequence omits numbers that are 2 mod 4. - Yuval Dekel, Jun 07 2003
Also note that 3 corresponds to Z[omega] (the Eisenstein integers) and 4 corresponds to Z[i] (the Gaussian integers).
Alaca & Williams cite Masley & Montgomery, saying the earlier authors "prove that there are precisely 29 distinct cyclotomic fields" with class number 1 (mentioning the n = 2 mod 4 caveat), and then give this sequence without the initial 1. - Alonso del Arte, Mar 10 2017
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REFERENCES
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Şaban Alaca & Kenneth S. Williams, Introductory Algebraic Number Theory. Cambridge: Cambridge University Press (2004): 343.
F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, p. 85, 1983.
J. Myron Masley, Where are the number fields with small class number?, pp. 221-242 of Number Theory Carbondale 1979, Lect. Notes Math. 751 (1982).
Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 259.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Alf van der Poorten, Notes on Fermat's Last Theorem, Wiley, 1996, p. 14.
L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 353.
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LINKS
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CROSSREFS
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KEYWORD
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fini,nonn,full,nice
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AUTHOR
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STATUS
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approved
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