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First repeated number in Sylvester's sequence modulo prime(n).
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%I #16 Apr 02 2015 04:10:38

%S 1,1,2,1,3,1,4,2,7,3,2,6,2,1,7,7,7,17,7,3,1,43,66,2,72,51,7,50,32,3,

%T 111,85,26,1,44,31,7,7,96,157,23,1,88,3,97,7

%N First repeated number in Sylvester's sequence modulo prime(n).

%C Sylvester's sequence (A000058) is an infinite coprime sequence, a fact that may lead to the incorrect intuition that all primes occur as factors of its terms. It's quite easy to check that no multiple of 5 occurs, since Sylvester's sequence modulo 5 is 2, 3, 2, 3, 2, 3, ...

%C If a multiple of p occurs in Sylvester's sequence at position j, then A000058(k) == 1 (mod p) for all k > j.

%C But if no multiple of p occurs in Sylvester's sequence at all, then Sylvester's sequence is fully periodic modulo p or it enters a cycle at some point.

%D J. J. Sylvester, Postscript to Note on a Point in Vulgar Fractions. American Journal of Mathematics Vol. 3, No. 4 (Dec., 1880): 388 - 389.

%e a(4) = 1, because the fourth prime is 7 and Sylvester's sequence modulo 7 is 2, 3, 0, 1, 1, 1, ...

%e a(5) = 3, because the fifth prime is 11 and Sylvester's sequence modulo 11 is 2, 3, 7, 10, 3, 7, 10, 3, 7, 10, ... (3 is the first number repeated).

%Y Cf. A007996, A126263, A255595.

%K nonn

%O 1,3

%A _Alonso del Arte_, Mar 16 2015