A000927 "First factor" (or relative class number) h- for cyclotomic field Q( exp(2 Pi / prime(n)) ). (Formerly M2711 N1088)

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, 2708534744692077051875131636

Offset

1,9

Comments

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

References

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).

Links

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

Formula

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)).

Example

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

Maple

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

Mathematica

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 *)

Prog

(PARI) { 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

Crossrefs

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.

Sequence in context: A223331 A101720 A093439 * A055513 A038226 A095866

Adjacent sequences: A000924 A000925 A000926 * A000928 A000929 A000930

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

Edited by Max Alekseyev, Oct 25 2012

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

Status

approved