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%I M2711 N1088 #59 Aug 09 2019 08:22:29
%S 1,1,1,1,1,1,1,1,3,8,9,37,121,211,695,4889,41241,76301,853513,3882809,
%T 11957417,100146415,838216959,13379363737,411322824001,3547404378125,
%U 9069094643165,63434933542623,161784800122409,1612072001362952,2604529186263992195,28496379729272136525,646901570175200968153,1753848916484925681747,687887859687174720123201,2333546653547742584439257,56234327700401832767069245,2708534744692077051875131636
%N "First factor" (or relative class number) h- for cyclotomic field Q( exp(2 Pi / prime(n)) ).
%C Washington gives a very extensive table. But beware errors: Washington incorrectly gives a(17) = 41421, a(25) = 411322842001 (corrected in the second edition).
%D Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, p. 429.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D L. C. Washington, Introduction to Cyclotomic Fields, Springer, pp. 353-360 (1st edition) pp. 412-420 (2nd edition).
%H Max Alekseyev, <a href="/A000927/b000927.txt">Table of n, a(n) for n = 1..100</a>
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/cn/index.htm">Factorizations of Cyclotomic Numbers</a>
%H M. Newman, <a href="http://www.jstor.org/stable/2004891">A table of the first factor for prime cyclotomic fields</a>, Math. Comp., 24 (1970), 215-219.
%H M. A. Shokrollahi, <a href="https://web.archive.org/web/20150911191857/http://algo.epfl.ch/~amin/TAB.html">Tables</a>
%F 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
%F a(n) = A061653(A000040(n)).
%e For n = 9, prime(9) = 23, a(9) = 3.
%e For n = 38, prime(38) = 163, a(38) = 2708534744692077051875131636.
%p f:= proc(n) uses LinearAlgebra;
%p local p,M;
%p p:= ithprime(n);
%p M:= Matrix((p-3)/2,(p-3)/2,(i,j) -> floor((i+1)*(j+2)/p) - floor(i*(j+2)/p));
%p abs(Determinant(M));
%p end proc:
%p 1, seq(f(n),n=3..50); # _Robert Israel_, Sep 20 2016
%t 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 *)
%o (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
%Y Subsequence of A061653.
%Y For the full class number h = h- * h+, see A055513, which agrees for the first 36 terms, assuming the Generalized Riemann Hypothesis.
%K nonn,nice
%O 1,9
%A _N. J. A. Sloane_
%E Edited by _Max Alekseyev_, Oct 25 2012
%E a(1)=1 prepended by _Max Alekseyev_, Mar 05 2018
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