A133154 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.

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, 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, 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

Offset: 1

Author

Alexander R. Povolotsky, Oct 08 2007

Keywords

• nonn

Comments

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.

Crossrefs

Cf. A131685.

Programs

PARI

{ 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 }
vector(100,n,a(prime(n))) /* Max Alekseyev */

Extensions

Definition simplified and comments edited by Jon E. Schoenfield, Nov 29 2013