%I #9 Nov 08 2017 10:48:48
%S 1,1,1,2,2,3,3,6,6,9,9,16,16,24,24,42,42,64,64,105,105,159,159,258,
%T 258,390,390,614,614,925,925,1441,1441,2162,2162,3317,3317,4951,4951,
%U 7526,7526,11191,11191,16841,16841,24923,24923,37253,37253,54912,54912,81493,81493,119629,119629,176549
%N Number of isomorphism classes of reduced Witt rings of fields with n orderings.
%C The number with 2m+1 orderings is the same as the number with 2m orderings (cf. A213331).
%H Thomas C. Craven, <a href="https://doi.org/10.1016/0097-3165(80)90006-0">An application of PĆ³lya's theory of counting to an enumeration problem arising in quadratic form theory</a>, J. Combin. Theory Ser. A 29 (1980), no. 2, 174--181. MR0583956 (81j:10027).
%p read transforms;
%p w:=proc(n) option remember; global did; local v; # did(n,d)=1 if d|n otherwise 0
%p if n=1 then 1 elif (n mod 2) = 1 then w(n-1);
%p else v:=n/2;
%p (1/n)* ( add(2*i*w(i)*did(v,i), i=1..v) +
%p add( add(2*i*w(i)*w(n-2*k)*did(k,i), i=1..k), k=1..v-1));
%p fi; end;
%p [seq(w(n),n=1..100)];
%Y Cf. A213331.
%K nonn
%O 1,4
%A _N. J. A. Sloane_, Jun 09 2012