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A000927
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"First factor" (or relative class number) h- for cyclotomic field Q( exp(2 Pi / prime(n)) ).
(Formerly M2711 N1088)
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8
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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
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OFFSET
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1,9
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COMMENTS
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Washington gives a very extensive table. But beware errors: Washington incorrectly gives a(17) = 41421, a(25) = 411322842001 (corrected in the second edition).
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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EXAMPLE
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For n = 9, prime(9) = 23, a(9) = 3.
For n = 38, prime(38) = 163, a(38) = 2708534744692077051875131636.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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For the full class number h = h- * h+, see A055513, which agrees for the first 36 terms, assuming the Generalized Riemann Hypothesis.
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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