A061653 -id:A061653 - 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

A000927 "First factor" (or relative class number) h- for cyclotomic field Q( exp(2 Pi / prime(n)) ).

Original entry on oeis.org

Offset: 1

N. J. A. Sloane

Washington gives a very extensive table. But beware errors: Washington incorrectly gives a(17) = 41421, a(25) = 411322842001 (corrected in the second edition).

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

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, p. 429.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
L. C. Washington, Introduction to Cyclotomic Fields, Springer, pp. 353-360 (1st edition) pp. 412-420 (2nd edition).

Max Alekseyev, Table of n, a(n) for n = 1..100
Hisanori Mishima, Factorizations of Cyclotomic Numbers
M. Newman, A table of the first factor for prime cyclotomic fields, Math. Comp., 24 (1970), 215-219.
M. A. Shokrollahi, Tables

Subsequence of A061653.

For the full class number h = h- * h+, see A055513, which agrees for the first 36 terms, assuming the Generalized Riemann Hypothesis.

f:= proc(n) uses LinearAlgebra;
  local p,M;
  p:= ithprime(n);
  M:= Matrix((p-3)/2,(p-3)/2,(i,j) -> floor((i+1)*(j+2)/p) - floor(i*(j+2)/p));
  abs(Determinant(M));
end proc:
1, seq(f(n),n=3..50); # Robert Israel, Sep 20 2016

a[n_]:= With[{p = Prime[n]}, If[n<4, 1, Abs[ Det[ Table[ Quotient[ (i+2)*(j+2), p] - Quotient[ (i+1)*(j+2), p], {i, 1, (p-1)/2-2}, {j, 1, (p-1)/2-2}]]]]]; Table[a[n], {n, 1, 38}] (* Jean-François Alcover, Aug 01 2013, translated from Pari; modified by G. C. Greubel, Aug 08 2019 *)

{ A000927(n) = if(n<3,return(1)); my(p=prime(n)); abs( matdet(matrix((p-1)/2-2, (p-1)/2-2, i, j, ((i+2)*(j+2))\p - ((i+1)*(j+2))\p)) ); } \\ Max Alekseyev, Oct 31 2012; corrected by G. C. Greubel and Michel Marcus, Aug 07 2019

For n>2, a(n) equals absolute value of determinant of the matrix with entries floor(i*j/p)-floor((i-1)*j/p), 3 <= i,j <= (p-1)/2, where p = prime(n) = A000040(n). - Max Alekseyev, Oct 31 2012

a(n) = A061653(A000040(n)).

Edited by Max Alekseyev, Oct 25 2012

a(1)=1 prepended by Max Alekseyev, Mar 05 2018

A005848 Cyclotomic fields with class number 1 (or with unique factorization).

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 48, 60, 84

Offset: 1

N. J. A. Sloane

Note that if n == 2 (mod 4) Q(zeta_n) is the same field as Q(zeta_{n/2}), so this sequence omits numbers that are 2 mod 4. - Yuval Dekel, Jun 07 2003
Also note that 3 corresponds to Z[omega] (the Eisenstein integers) and 4 corresponds to Z[i] (the Gaussian integers).
Alaca & Williams cite Masley & Montgomery, saying the earlier authors "prove that there are precisely 29 distinct cyclotomic fields" with class number 1 (mentioning the n = 2 mod 4 caveat), and then give this sequence without the initial 1. - Alonso del Arte, Mar 10 2017

Şaban Alaca & Kenneth S. Williams, Introductory Algebraic Number Theory. Cambridge: Cambridge University Press (2004): 343.
F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, p. 85, 1983.
J. Myron Masley, Where are the number fields with small class number?, pp. 221-242 of Number Theory Carbondale 1979, Lect. Notes Math. 751 (1982).
Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 259.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Alf van der Poorten, Notes on Fermat's Last Theorem, Wiley, 1996, p. 14.
L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 353.

Michael Baake and Uwe Grimm, A note on shelling, arXiv:math/0203025 [math.MG], 2002-2003.
E. Bugarin, M. de las Peñas, and D. Frettlöh, Perfect colourings of cyclotomic integers, arXiv:0905.4048 [math.GR], 2009-2012.
Hendrik W. Lenstra and A. J. van der Poorten, Euclidean number fields 1, Math. Intelligencer 2 (1979): pp. 6-15.
J. Myron Masley and Hugh L. Montgomery, Cyclotomic fields with unique factorization, Journal für die reine und angewandte Mathematik 286/287 (1976), 248-256.

Cf. A061653.

A035115 Relative class number h- of cyclotomic field Q(zeta_m) where m is n-th term of A035113.

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, 1, 1, 1, 1, 1, 3, 1, 2, 1, 3, 2, 3, 1, 8, 9, 5, 17, 8, 5, 9, 4, 37, 9, 7, 19, 19, 121, 10, 11, 55, 55, 11, 211, 43, 69, 201, 695, 64, 351, 13, 468, 39, 507, 156, 84, 75, 4889, 2593, 1536, 10752, 41241, 76301, 1280, 6795

Offset: 1

N. J. A. Sloane

L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 353.

Cf. A000010, A035113, A035114, A055513, A061494, A061653.

(* This is only a recomputation of the existing data. *)
terms = 76;
A035113 = Sort[Select[Range[3 terms], Mod[#, 4] != 2&], EulerPhi[#1] <= EulerPhi[#2]&];
A061653 = Import["https://oeis.org/A061653/b061653.txt", "Table"][[All, 2]];
a[n_] := A061653[[A035113[[n]] ]];
Array[a, terms] (* Jean-François Alcover, Aug 17 2019 *)

a(n) = A061653(A035113(n)).

Changed 41421 to 41241, value is given incorrectly in first edition of Washington, Matthew Johnson, Jul 20 2013

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].

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 16 2001

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

			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.

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

Cf. A035115, A055513, A061653, A042965.

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

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

cp's OEIS Frontend

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

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A000927 "First factor" (or relative class number) h- for cyclotomic field Q( exp(2 Pi / prime(n)) ).

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A005848 Cyclotomic fields with class number 1 (or with unique factorization).

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A035115 Relative class number h- of cyclotomic field Q(zeta_m) where m is n-th term of A035113.

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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].

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