A334048 Primes p that set a new record for the number of bases 1 < b < p for which p is a base-b Wieferich prime and b is not a perfect power.

2, 11, 71, 269, 653, 5107, 103291, 728471, 2544079

Offset: 1

Author

Jeppe Stig Nielsen, Sep 06 2020

Keywords

• nonn
• more

Comments

p is a base-b Wieferich prime iff b^(p-1) == 1 (mod p^2).

Records in A248865 sometimes arise when all the b values (bases) are powers of the same small integer. By excluding powers, we find primes that are Wieferich in many "independent" ways.

Examples

			Prime 5107 is Wieferich to six bases (under 5107), namely 560, 1209, 1779, 2621, 4295, 4361, none of which are perfect powers. A prime such as 1093 is Wieferich to ten bases, namely 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024; however, when dismissing perfect powers, only one of the ten bases is left. In fact, no prime less than 5107 has six or more bases when perfect powers are dismissed, so 5107 sets a record and is included in this sequence.
		

Crossrefs

Cf. A134307, A175932, A242830, A248865.

Programs

PARI

r=-oo; forprime(p=2,, i=sum(b=2,p-1,!ispower(b) && Mod(b,p^2)^(p-1)==1); if(i>r, print1(p, ", "); r=i))

Extensions

a(8)-a(9) from Kellen Shenton added by Jeppe Stig Nielsen, Sep 12 2020