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A256147
First repeated number in Sylvester's sequence modulo prime(n).
0
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, 111, 85, 26, 1, 44, 31, 7, 7, 96, 157, 23, 1, 88, 3, 97, 7
OFFSET
1,3
COMMENTS
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, ...
If a multiple of p occurs in Sylvester's sequence at position j, then A000058(k) == 1 (mod p) for all k > j.
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.
REFERENCES
J. J. Sylvester, Postscript to Note on a Point in Vulgar Fractions. American Journal of Mathematics Vol. 3, No. 4 (Dec., 1880): 388 - 389.
EXAMPLE
a(4) = 1, because the fourth prime is 7 and Sylvester's sequence modulo 7 is 2, 3, 0, 1, 1, 1, ...
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).
CROSSREFS
KEYWORD
nonn
AUTHOR
Alonso del Arte, Mar 16 2015
STATUS
approved