A045976 - OEIS

%I #21 Mar 08 2022 08:17:41

%S 1,1,0,9,121,2704,118336,4092529,0,97734390625,69057113823721,

%T 61675735716000000,120510849036513735801,204044070747203744272384,0,

%U 12086640655039701533948077217761,424809992547069296975659357323606889

%N A product related to n-th roots of 1.

%C a(3*(2*k+1)) = 0 for all k because there are some 3rd roots involved.

%D Related works addressing "the first case" of Fermat's Theorem as studied by mathematicians at the end of the 19th century:

%D L. D. Dickson, History of the Theory of Numbers, Vol II, page 756, lines -5 to -1, and page 757, lines 1 to 4: Some related work of E. Wendt is described.

%D L. D. Dickson, History of the Theory of Numbers, Vol II, page 757, lines 14 to 24: Some related work of G. B. Mathews (in 1895) is described.

%D R. Nogues, Théorème de Fermat Son Histoire (in French), reprint (1992) Jacques Gabay's edition of the original of 1932, page 132, lines 1 to 13 for G. B. Mathews's work, cited in the second Dickson reference above.

%F a(n) = Product_{u^n=1, u != 1} ((1+u)^n + 1).

%o (PARI) a(n)=sum(k=1,n-1,(1+exp(2*Pi*I/n))^n+1) \\ _Charles R Greathouse IV_, May 13 2011

%K nonn

%O 1,4

%A _Luis H. Gallardo_