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A003640 Number of genera of imaginary quadratic field with discriminant -k, k = A003657(n).
(Formerly M0090)
6

%I M0090

%S 1,1,1,1,1,2,1,2,1,2,1,2,2,2,1,1,2,2,2,2,1,1,2,1,1,1,4,2,2,2,2,1,2,1,

%T 2,2,2,2,4,2,1,1,4,2,1,2,2,1,2,2,2,1,2,1,4,1,2,2,2,1,4,1,2,1,2,2,2,1,

%U 1,4,4,2,2,1,2,2,2,1,4,2,4,1,4,2,1,4,4,1,2,2,2,2,2,2,2,1,4,1,4,2,2,2,2,1,2

%N Number of genera of imaginary quadratic field with discriminant -k, k = A003657(n).

%C The number of genera of a quadratic field is equal to the number of elements x in the class group such that x^2 = e where e is the identity. - _Jianing Song_, Jul 24 2018

%D D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Andrew Howroyd, <a href="/A003640/b003640.txt">Table of n, a(n) for n = 1..1000</a>

%H Rick L. Shepherd, <a href="https://libres.uncg.edu/ir/uncg/listing.aspx?id=15057">Binary quadratic forms and genus theory</a>, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

%H <a href="/index/Qua#quadfield">Index entries for sequences related to quadratic fields</a>

%F a(n) = 2^(t-1) where t = number of distinct prime divisors of A003657(n).

%F a(n) = 2^(omega(A003657(n)) - 1).

%t okQ[n_] := n != 1 && SquareFreeQ[n/2^IntegerExponent[n, 2]] && (Mod[n, 4] == 3 || Mod[n, 16] == 8 || Mod[n, 16] == 4);

%t Reap[For[n = 1, n <= 1000, n++, If[okQ[n], Sow[2^(PrimeNu[n]-1)]]]][[2, 1]] (* _Jean-Fran├žois Alcover_, Aug 16 2019, after _Andrew Howroyd_ *)

%o (PARI) for(n=1, 1000, if(isfundamental(-n), print1(2^(omega(n) - 1), ", "))) \\ _Andrew Howroyd_, Jul 24 2018

%o (PARI) for(n=1, 1000, if(isfundamental(-n), print1(2^#select(t->t%2==0, quadclassunit(-n).cyc), ", "))) \\ _Andrew Howroyd_, Jul 24 2018

%Y Cf. A001221 (omega), A003641, A003642, A003643, A003657, A191408.

%K nonn,easy,nice

%O 1,6

%A _N. J. A. Sloane_, _Mira Bernstein_

%E Name clarified and offset corrected by _Jianing Song_, Jul 24 2018

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Last modified January 18 16:27 EST 2020. Contains 331011 sequences. (Running on oeis4.)