A055513 Class number h = h- * h+ of cyclotomic field Q( exp(2 Pi / prime(n)) ).

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

Offset: 1

Author

N. J. A. Sloane, Jun 16 2001

Keywords

• nonn
• nice

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)

Examples

			For n = 9, prime(9) = 23, a(9) = 3.
For n = 38, prime(38) = 163, a(38) = 4*2708534744692077051875131636 = 10834138978768308207500526544.
		

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)
• Hisanori Mishima, Factorizations of Cyclotomic Numbers
• 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

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.

Cf. A035115, A055514, A061653.

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