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%I #10 Nov 29 2013 21:19:13
%S 0,0,0,2,7,11,2,5,3,8,5,26,2,2,9,16,6,14,9,9,3,10,3,10,4,2,5,2,13,2,3,
%T 2,3,21,8,22,2,3,2,5,5,2,3,2,4,2,2,7,44,7,16,3,4,3,2,19,22,3,3,26,7,
%U 16,12,2,9,6,2,14,3,4,9,6,4,19,15,6,4,6,16,5,11,9,5,4,2,3,18,3,7,9,18,16,3,8
%N a(n) is the smallest m<=p-1 such that p-1 is the only value of j in 1<=j<=2p for which m^j+j==0 (mod p), where p is the n-th prime.
%C Andrew Granville, based on submitter's analysis of the data in A131685, made the following conjecture: "For some n with 1<=n<=p-1, there does not exist a value of j, with 1 <= j <= 2p, other than j=p-1, for which n^j+j == 0 (mod p)." Max Alekseyev's calculations confirm that the conjecture is true for the primes between 5 and 10^5. The sequence consists of the first such "n" (referred to as "m" in this sequence's definition) for each prime. a(n)=0 means that there is no corresponding m; this occurs at n=1 (p=2), n=2 (p=3), and n=3 (p=5), and at no other primes p<10^5.
%o { a(p) = for(n=1,p-1, local(j=1); while(j<=2*p, if( j!=p-1 && Mod(n,p)^j==-j, break); j++); if(j>2*p,return(n)); ); 0 }
%o vector(100,n,a(prime(n))) /* _Max Alekseyev_ */
%Y Cf. A131685.
%K nonn
%O 1,4
%A _Alexander R. Povolotsky_, Oct 08 2007
%E Definition simplified and comments edited by _Jon E. Schoenfield_, Nov 29 2013
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