A274175 Composite numbers c that set a new record for the number of bases b with 1 < b < c such that c satisfies b^(c-1) == 1 (mod c^2), i.e., such that c is a base-b "Wieferich pseudoprime".

133, 1065, 141373

Offset: 1

Author

Felix Fröhlich, Jun 12 2016

Keywords

• nonn
• hard
• more
• bref

Comments

a(4) > 253263 if it exists.

Is the sequence infinite?

Let x be the integer sequence defined as x(n) = number of bases 1 < b < c such that c is a base-b "Wieferich pseudoprime", where c is the n-th composite number (that sequence does not have its own entry in the OEIS). Then a(n) is the sequence of composites where x(n) reaches record values.

Let y be the integer sequence defined as y(n) = smallest composite c with exactly n bases 1 < b < c such that c is a base-b "Wieferich pseudoprime". Is a(n) = y(n) for all n?

For every b with 1 < b < c such that c is a base-b "Wieferich pseudoprime", every prime factor p of c is a base-b Wieferich prime.

Examples

			c = 141373 satisfies b^(c-1) == 1 (mod c^2) for three values of b with 1 < b < c, namely b = 23382, 36620 and 130595. Since no other composite c < 141373 exists that has more than two such b, 141373 is a term of the sequence.
		

Crossrefs

Cf. A194946, A248865, A256517, A267288.

Programs

PARI

my(r=0, i); forcomposite(c=1, , i=0; for(b=2, c-1, if(Mod(b, c^2)^(c-1)==1, i++)); if(i > r, print1(c, ", "); r=i))