A061653 Relative class number h- of cyclotomic field Q(zeta_n).

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

Offset: 1

Author

N. J. A. Sloane, Jun 16 2001

Keywords

• nonn
• easy
• nice

Comments

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

From Richard N. Smith, Jul 15 2019: (Start)

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

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

Examples

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

Links

• Richard N. Smith, Table of n, a(n) for n = 1..256 (terms 1..163 from R. J. Mathar)
• Dylan Johnston, Diego Martín Duro, and Dmitriy Rumynin, Disconnected Reductive Groups: Classification and Representations, arXiv:2409.06375 [math.RT], 2024. See p. 14.
• L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 353. [WARNING: The table contains errors for n=59, 97, ...]

Crossrefs

Contains A000927, A035115, A061494 as subsequences.

Cf. A055513, A005848.

Formula

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

Extensions

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

Missing term a(1) = 1 inserted by N. J. A. Sloane, Feb 05 2009 at the suggestion of Tanya Khovanova

More terms from R. J. Mathar, Feb 06 2009

a(59) changed from 41421 to 41241 (given correctly in 2nd edition of Washington), Matthew Johnson, Jul 20 2013

a(59) in b-file changed as above by Andrew Howroyd, Feb 23 2018

a(97) corrected, a(163) added by Max Alekseyev, Mar 05 2018