A335507 - OEIS

%I #8 Jul 05 2020 09:03:08

%S 3,2,4,3,5,28,8,9,11,7,5,9,20,14,23,13,29,15,11,35,9,13,41,11,32,25,

%T 17,53,27,28,7,13,17,23,37,15,39,27,83,43,89,45,19,32,28,11,21,37,113,

%U 19,29,34,40,25,16,131,67,15,69,35,47,73,17,31,39,79,33,21,173,29,32,179

%N Index of the least Wendt determinant (A048954) divisible by prime(n).

%C It has been conjectured by Michael B Rees that there exists for every prime a Wendt determinant divisible by that prime. However the conjecture has been proved for all prime divisors equivalent to -1 (mod 6) - (see Lehmer link below).

%H Charles Helou, <a href="http://dx.doi.org/10.1090/S0025-5718-97-00870-3">On Wendt's Determinant</a>, Math. Comp., 66 (1997) No. 219, 1341-1346.

%H Emma Lehmer, <a href="https://doi.org/10.1090/S0002-9904-1935-06210-X">On a Resultant Connected with Fermat's Last Theorem</a>, Bull. Amer. Math. Soc. 41 (1935), 864-867.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CirculantMatrix.html">Circulant matrix</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Circulant_matrix">Circulant matrix</a>.

%e a(5) = 5 because Wendt(5) = 3751 = 11^2*131. It is divisible by prime(5) = 11 and Wendt(5) is the least Wendt determinant divisible by 11.

%t Wendt[n_]:=Module[{x},Resultant[x^n-1,(1+x)^n-1,x]];

%t findW[n_]:= Module[{m=1},While[!IntegerQ[Wendt[m]/n]||Mod[m,6]==0,m++];m];

%t Table[findW[Prime[n]],{n,1,100}]

%Y Cf. A048954.

%K nonn

%O 1,1

%A _Frank M Jackson_ and Michael B Rees, Jun 11 2020