A061653 - OEIS

%I #50 Sep 16 2024 12:51:53

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,8,1,9,1,1,1,

%T 1,1,37,1,2,1,121,1,211,1,1,3,695,1,43,1,5,3,4889,1,10,2,9,8,41241,1,

%U 76301,9,7,17,64,1,853513,8,69,1,3882809,3,11957417,37,11,19,1280,2,100146415

%N Relative class number h- of cyclotomic field Q(zeta_n).

%C Note that if n == 2 (mod 4), Q(zeta_n) is the same field as Q(zeta_{n/2}).

%C From _Richard N. Smith_, Jul 15 2019: (Start)

%C For prime p, p divides a(p) (or a(2p)) if and only if p is in A000928.

%C For prime p, p divides a(4p) if and only if p is in A250216. (End)

%H Richard N. Smith, <a href="/A061653/b061653.txt">Table of n, a(n) for n = 1..256</a> (terms 1..163 from R. J. Mathar)

%H Dylan Johnston, Diego Martín Duro, and Dmitriy Rumynin, <a href="https://arxiv.org/abs/2409.06375">Disconnected Reductive Groups: Classification and Representations</a>, arXiv:2409.06375 [math.RT], 2024. See p. 14.

%H L. C. Washington, <a href="https://link.springer.com/content/pdf/bbm%3A978-1-4684-0133-2%2F1.pdf">Introduction to Cyclotomic Fields</a>, Springer, p. 353. [WARNING: The table contains errors for n=59, 97, ...]

%F For prime p, a(p) = A000927(A000720(p)).

%e Q(zeta_23) = 3 is the first time that h- is bigger than 1.

%Y Contains A000927, A035115, A061494 as subsequences.

%Y Cf. A055513, A005848.

%K nonn,easy,nice

%O 1,23

%A _N. J. A. Sloane_, Jun 16 2001

%E Washington gives an extensive table on pp. 353-360.

%E Missing term a(1) = 1 inserted by _N. J. A. Sloane_, Feb 05 2009 at the suggestion of _Tanya Khovanova_

%E More terms from _R. J. Mathar_, Feb 06 2009

%E a(59) changed from 41421 to 41241 (given correctly in 2nd edition of Washington), _Matthew Johnson_, Jul 20 2013

%E a(59) in b-file changed as above by _Andrew Howroyd_, Feb 23 2018

%E a(97) corrected, a(163) added by _Max Alekseyev_, Mar 05 2018