A187929 - OEIS

%I #40 Mar 17 2024 02:15:43

%S 1,3,5,7,11,13,15,17,19,23,27,29,31,35,37,39,41,43,47,51,53,55,59,61,

%T 63,67,71,73,75,77,79,83,85,87,89,91,95,97,99,101,103,107,109,111,113,

%U 115,119,123,125,127,131,135,137,139,143,147,149,151,155,157,159

%N Odd numbers k such that 1^((k-1)/2) + 2^((k-1)/2) + .... + (k-1)^((k-1)/2) == 0 (mod k).

%C Asymptotic density is 0.379...

%H Amiram Eldar, <a href="/A187929/b187929.txt">Table of n, a(n) for n = 1..10000</a>

%H José María Grau, Florian Luca and Antonio M. Oller-Marcén, <a href="https://doi.org/10.1007/s10114-011-1148-7">On a variant of Giuga numbers</a>, Acta Mathematica Sinica, English Series, Vol. 28 No. 4 (2012), pp 653-660; <a href="http://arxiv.org/abs/1103.3428">preprint</a>, arXiv:1103.3428 [math.NT], 2011.

%t gi[n_]:=Mod[Sum[PowerMod[j,(n-1)/2,n],{j,n-1}],n]; Select[ Range[1,300,2], gi[#]==0&]

%o (PARI) is(n)=my(e=(n-1)/2);sum(k=1,n-1,Mod(k,n)^e)==0;

%o select(is,vector(1000,i,2*i-1)) \\ on older versions, switch the arguments

%o \\ _Charles R Greathouse IV_, Mar 19 2011

%Y Cf. A007850.

%K nonn

%O 1,2

%A _José María Grau Ribas_, Mar 16 2011