A000927 "First factor" (or relative class number) h- for 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, 2708534744692077051875131636

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

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

Examples

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

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

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.

Programs

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 *)

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

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)).

Extensions

Edited by Max Alekseyev, Oct 25 2012

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