A256147 First repeated number in Sylvester's sequence modulo prime(n).

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.

Links

Table of n, a(n) for n=1..46.

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

Cf. A007996, A126263, A255595.

Sequence in context: A029171 A354185 A322994 * A079786 A032451 A214494

Adjacent sequences: A256144 A256145 A256146 * A256148 A256149 A256150

Keyword

nonn

Author

Alonso del Arte, Mar 16 2015

Status

approved