A268310 Largest k > 0 such that a base b with 1 < b < c exists such that b^(c-1) == 1 (mod c^k), where c is the n-th composite number, or 0 if no such k exists.

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0

Offset

1

Comments

a(n) > 0 iff c is a term of A039769.

a(n) > 1 iff c is a term of A267288. In that case, c can be called a "base-b Wieferich pseudoprime", which first happens for n = 100.

Does an integer constant t exist such that a(n) <= t for all n?

Links

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Example

The largest k such that c = 133, i.e. A002808(100), satisfies the congruence b^(c-1) == 1 (mod c^k) for 1 < b < 133 is 2, which happens for b = 68. Since there is no other b with 1 < b < c and c satisfying this congruence for a larger k, a(100) = 2.

Prog

(PARI) forcomposite(c=1, 1e2, my(maxexp=0, k=1); for(b=2, c-1, while(Mod(b, c^k)^(c-1)==1, k++); if(k-1 > maxexp, maxexp=k-1)); print1(maxexp, ", "))

Crossrefs

Cf. A039769, A244752, A254444, A256517, A267288.

Sequence in context: A331169 A144602 A275855 * A283316 A284508 A347519

Adjacent sequences: A268307 A268308 A268309 * A268311 A268312 A268313

Keyword

nonn

Author

Felix Fröhlich, Jan 31 2016

Status

approved