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A055513 Class number h = h- * h+ of cyclotomic field Q( exp(2 Pi / prime(n)) ). 9
1, 1, 1, 1, 1, 1, 1, 1, 3, 8, 9, 37, 121, 211, 695, 4889, 41241, 76301, 853513, 3882809, 11957417, 100146415, 838216959, 13379363737, 411322824001, 3547404378125, 9069094643165, 63434933542623, 161784800122409, 1612072001362952, 2604529186263992195, 28496379729272136525, 646901570175200968153, 1753848916484925681747, 687887859687174720123201, 2333546653547742584439257, 56234327700401832767069245, 10834138978768308207500526544 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,9
COMMENTS
Washington gives a very extensive table (but beware errors!).
From Jianing Song, Nov 10 2023: (Start)
h+(n) denotes the class number of Q(exp(2*Pi/n) + exp(-2*Pi/n)).
Primes p such that h+(p) != 1 are listed in A230869. As a result, if prime(n) is not in A230869, then a(n) = A000927(n), otherwise a(n) = A000927(n) * A230870(m) for prime(n) = A230869(m). (End)
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, p. 429.
L. C. Washington, Introduction to Cyclotomic Fields, Springer, pp. 353-360.
LINKS
Jianing Song, Table of n, a(n) for n = 1..100 (b-file based on data of A000927, A230869 and A230870)
M. Newman, A table of the first factor for prime cyclotomic fields, Math. Comp., 24 (1970), 215-219.
Rene Schoof, Class numbers of real cyclotomic fields of prime conductor, Math. Comp., 72 (2002), 913-937.
M. A. Shokrollahi, Tables
EXAMPLE
For n = 9, prime(9) = 23, a(9) = 3.
For n = 38, prime(38) = 163, a(38) = 4*2708534744692077051875131636 = 10834138978768308207500526544.
CROSSREFS
For the relative class number h-, see A000927, which agrees for the first 36 terms, assuming the Generalized Riemann Hypothesis. See also A230869 and A230870.
Sequence in context: A101720 A093439 A000927 * A038226 A095866 A067569
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane, Jun 16 2001
EXTENSIONS
Washington incorrectly gives a(17) = 41421, a(25) = 411322842001.
Edited by Max Alekseyev, Oct 25 2012
a(1) = 1 prepended by Jianing Song, Nov 10 2023
STATUS
approved

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)