A035115 -id:A035115

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A055513

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

+10
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)

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

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.

Cf. A035115, A055514, A061653.

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

A061653

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,23

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)

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, ...]

FORMULA

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

EXAMPLE

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

CROSSREFS

Contains A000927, A035115, A061494 as subsequences.

Cf. A055513, A005848.

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Jun 16 2001

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

STATUS

approved

A061494

Relative class number h- of cyclotomic field Q(zeta_n) where n runs through positive integers not congruent to 2 (mod 4) [A042965, but omitting the initial 0].

+10
2

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 8, 9, 1, 1, 1, 1, 37, 2, 1, 121, 211, 1, 1, 695, 1, 43, 5, 3, 4889, 10, 2, 9, 41241, 1, 76301, 7, 17, 64, 853513, 8, 69, 3882809, 3, 11957417, 11, 19, 1280, 100146415, 5, 2593, 838216959, 1, 6205, 1536, 55, 13379363737, 53872

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,17

COMMENTS

First edition of Washington incorrectly gives a(44) = h-(Q(zeta_59)) = 41421. [Matthew Johnson, Jul 20 2013]

REFERENCES

L. C. Washington, Introduction to Cyclotomic Fields, Springer, pp. 353-360.

LINKS

Table of n, a(n) for n=1..68.

FORMULA

a(n) = A061653(A042965(n+1)). - M. F. Hasler, Feb 04 2009

EXAMPLE

n=17: the 17th number not == 2 mod 4 is 23, and Q(zeta_23) = 3 is the first time that h- is bigger than 1, so a(17) = 3.

CROSSREFS

Cf. A035115, A055513, A061653, A042965.

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, Jun 16 2001

EXTENSIONS

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

More terms (from b-file of A061653), Joerg Arndt, Oct 07 2012

a(44) corrected by Matthew Johnson, Jul 20 2013

STATUS

approved

A230869

Primes p such that the class number h-tilde_p^{+} of the real cyclotomic field Q(zeta_p + zeta_p^(-1)) is greater than 1.

+10
2

163, 191, 229, 257, 277, 313, 349, 397, 401, 457, 491, 521, 547, 577, 607, 631, 641, 709, 733, 761, 821, 827, 829, 853, 857, 877, 937, 941, 953, 977, 1009, 1063, 1069, 1093, 1129, 1153, 1229, 1231, 1297, 1373, 1381, 1399, 1429, 1459, 1489, 1567, 1601, 1697, 1699, 1777, 1789, 1831, 1861, 1873, 1879, 1889, 1901, 1951

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Taken from the "Main Table" of Schoof.

There is a very slight chance that some primes are missing. In the unlikely event that the number that Schoof calls h-tilde_p is 1, while the actual class number h_p is actually not equal to 1, the prime p would be missing (see the Schoof and Miller articles for details).

LINKS

Table of n, a(n) for n=1..58.

John C. Miller, Class numbers of totally real fields and applications to the Weber class number problem, arXiv:1405.1094 [math.NT], 2014.

John C. Miller, Real cyclotomic fields of prime conductor and their class numbers, arXiv:1407.2373 [math.NT], 2014.

Rene Schoof, Class numbers of real cyclotomic fields of prime conductor, Math. Comp., 72 (2002), 913-937.

CROSSREFS

Cf. A230870 (for the actual class numbers).

See also A000927, A055513, A035113, A035114, A035115.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nov 06 2013

STATUS

approved

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