A000927 -id:A000927 - cp's OEIS frontend

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

N. J. A. Sloane, Jun 16 2001

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)

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

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.

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

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.

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

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

Original entry on oeis.org

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

N. J. A. Sloane, Jun 16 2001

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)

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

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, ...]

Contains A000927, A035115, A061494 as subsequences.

Cf. A055513, A005848.

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

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

A088922 Consider the n X n matrix with entries (i*j mod n), where i,j=0..n-1; a(n) = rank of this matrix over the real numbers.

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 4, 6, 6, 7, 6, 10, 7, 9, 10, 11, 9, 13, 10, 14, 13, 13, 12, 18, 14, 15, 16, 18, 15, 21, 16, 20, 19, 19, 20, 25, 19, 21, 22, 26, 21, 27, 22, 26, 27, 25, 24, 32, 26, 29, 28, 30, 27, 33, 30, 34, 31, 31, 30, 40, 31, 33, 36, 37, 35, 39, 34, 38, 37, 41, 36, 46, 37, 39, 42, 42, 41, 45, 40, 48, 44, 43, 42, 52, 45, 45, 46, 50, 45, 55, 48, 50, 49, 49, 50, 58, 49, 53, 54, 57

Offset: 1

Max Alekseyev, Dec 01 2003

nonn

Possibly related to Maillet's determinants.

			From _Alexander Adam_, Nov 10 2012: (Start)
a(2^m) = 2^(m-1) + m - 1.
Let p >= 3 be a prime number. Then a(p^m) = (p^m + 1) / 2 + m - 1.
a(625000) = a(2^3*5^7) = 2^2*5^7 + 4 * 8 - 2 = 312530. (End)

Alexander Adam, Proof of the formula
Wikipedia, Maillet's determinant

Cf. A218322, A055513, A203411, A000927.

a[n_] := MatrixRank[Table[Table[Mod[i * j, n], {j, 0, n - 1}], {i, 0, n - 1}]]; Array[a,100] (* Alexander Adam, Nov 10 2012 *)

PARI
```
a(n) = matrank(matrix(n,n,i,j,(i*j)%n))
```

Let n = Prod_{i>0} p_i^{m_i} be the prime factorization of n. Then a(n) = floor((n + 1)/2) + Prod_{i>0} (m_i + 1) - 2. - Alexander Adam, Nov 10 2012

a(n) = A000005(n) + A110654(n) - 2.

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.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 06 2013

nonn

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).

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.

Cf. A230870 (for the actual class numbers).

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

A193179 Number of classes of two-dimensional generalized Bravais lattices with 2n-fold symmetry.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 2, 9, 1, 9, 17, 1, 32, 1, 3, 34, 19, 2, 5, 121, 1, 211, 55, 1, 201, 695, 32, 43

Offset: 2

N. J. A. Sloane, Jul 17 2011

nonn

Related to class numbers of cyclotomic fields, although the precise relationship is not clear to me. Compare A000927.

Mermin, N. David; Rokhsar, Daniel S.; and Wright, David C.; Beware of 46-fold symmetry: the classification of two-dimensional quasicrystallographic lattices. Phys. Rev. Lett. 58 (1987), 2099-2101.

See A193175 for values of 2n such that a(n) = 1.

Cf. A000927.

A218322 Maillet determinant for prime(n).

Original entry on oeis.org

1, -5, 49, 14641, -371293, -410338673, 16983563041, 124279533640947, -82085029703668817512, 6812495416987166882889, -16890053810563300749953435929, -531714676529925182191868570093681, 98548851401030959947062957685234211, 4247541973383735863138308138153477847255, -62534081783371829558502906501683809565833328077, 1581923629964045589238110056212521488781448927448053161

Offset: 2

Max Alekseyev, Oct 25 2012

sign

Wikipedia, Maillet's determinant

Cf. A055513, A203411, A000927, A088922.

a(n) = p=prime(n); matdet(matrix((p-1)/2,(p-1)/2,i,j,(i/j)%p))

a(n) = (-p)^((p-3)/2) * h^-(p) = (-1)^((p-3)/2) * A203411(n) * A000927(n), where p=A000040(n).

A309520 Primes p for which h1(p)/G(p) has a record value.

Original entry on oeis.org

3, 5, 7, 11, 23, 73, 89, 179, 233, 761, 1451, 2741, 4391, 5231, 42611, 198221, 305741, 6766811, 1326662801, 1979990861, 4735703723, 9697282541, 35285447111, 45169368641, 169684421321, 187946428721

Offset: 1

Michel Marcus, Aug 06 2019

h1 is given by A000927, and G(p) = 2*p*(p/(4*Pi^2))^((p-1)/4).

a(1)-a(12) from Fung et al., a(13)-a(14) from Shokrollahi, a(15)-a(17) from Broadhurst, a(18) from Languasco et al. and Broadhurst, a(19)-a(26) from Broadhurst.

David Broadhurst, Kummer ratio champions with p < 10^12, July 22 2021.
Gilbert Fung, Andrew Granville and Hugh C. Williams, Computation of the first factor of the class number of cyclotomic fields, Journal of Number Theory, Volume 42, Issue 3, November 1992, Pages 297-312. See Table IV A p. 304.
Alessandro Languasco, Pieter Moree, Sumaia Saad Eddin and Alisa Sedunova, Computation of the Kummer ratio of the class number for prime cyclotomic fields, arXiv:1908.01152 [math.NT], 2019.
M. A. Shokrollahi, Relative class number of imaginary Abelian fields of prime conductor below 10000, Math. Comp. 68 (1999), 1717-1728. See Table 4 p. 1726.

Cf. A000927 (h1), A073010 (value for p=3).

h1(p) = if (p<5, 1, abs( matdet(matrix((p-1)/2-2, (p-1)/2-2, i, j, ((i+2)*(j+2))\p - ((i+1)*(j+2))\p)) )); \\ A000927
G(p) = 2*p*(p/(4*Pi^2))^((p-1)/4);
lista(nn) = {my(m = 0, nm); for (n=2, nn, p = prime(n); if ((nm = h1(p)/G(p)) > m, print1(p, ", "); m = nm););}

Missing terms 42611, 198221, 305741 and terms larger than 6766811 added by Alessandro Languasco on behalf of David Broadhurst, Jul 24 2021

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A088922 Consider the n X n matrix with entries (i*j mod n), where i,j=0..n-1; a(n) = rank of this matrix over the real numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

A193179 Number of classes of two-dimensional generalized Bravais lattices with 2n-fold symmetry.

Views

Author

Keywords

Comments

References

Crossrefs

A218322 Maillet determinant for prime(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A309520 Primes p for which h1(p)/G(p) has a record value.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions