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

%I A055513 #26 Nov 25 2023 11:54:38
%S A055513 1,1,1,1,1,1,1,1,3,8,9,37,121,211,695,4889,41241,76301,853513,3882809,
%T A055513 11957417,100146415,838216959,13379363737,411322824001,3547404378125,
%U A055513 9069094643165,63434933542623,161784800122409,1612072001362952,2604529186263992195,28496379729272136525,646901570175200968153,1753848916484925681747,687887859687174720123201,2333546653547742584439257,56234327700401832767069245,10834138978768308207500526544
%N A055513 Class number h = h- * h+ of cyclotomic field Q( exp(2 Pi / prime(n)) ).
%C A055513 Washington gives a very extensive table (but beware errors!).
%C A055513 From _Jianing Song_, Nov 10 2023: (Start)
%C A055513 h+(n) denotes the class number of Q(exp(2*Pi/n) + exp(-2*Pi/n)).
%C A055513 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)
%D A055513 Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, p. 429.
%D A055513 L. C. Washington, Introduction to Cyclotomic Fields, Springer, pp. 353-360.
%H A055513 Jianing Song, <a href="/A055513/b055513.txt">Table of n, a(n) for n = 1..100</a> (b-file based on data of A000927, A230869 and A230870)
%H A055513 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/cn/index.htm">Factorizations of Cyclotomic Numbers</a>
%H A055513 M. Newman, <a href="https://doi.org/10.1090/S0025-5718-1970-0257029-5">A table of the first factor for prime cyclotomic fields</a>, Math. Comp., 24 (1970), 215-219.
%H A055513 Rene Schoof, <a href="https://doi.org/10.1090/S0025-5718-02-01432-1">Class numbers of real cyclotomic fields of prime conductor</a>, Math. Comp., 72 (2002), 913-937.
%H A055513 M. A. Shokrollahi, <a href="http://algo.epfl.ch/~amin/TAB.html">Tables</a>
%e A055513 For n = 9, prime(9) = 23, a(9) = 3.
%e A055513 For n = 38, prime(38) = 163, a(38) = 4*2708534744692077051875131636 = 10834138978768308207500526544.
%Y A055513 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.
%Y A055513 Cf. A035115, A055514, A061653.
%K A055513 nonn,nice
%O A055513 1,9
%A A055513 _N. J. A. Sloane_, Jun 16 2001
%E A055513 Washington incorrectly gives a(17) = 41421, a(25) = 411322842001.
%E A055513 Edited by _Max Alekseyev_, Oct 25 2012
%E A055513 a(1) = 1 prepended by _Jianing Song_, Nov 10 2023