A242830 -id:A242830 - cp's OEIS frontend

A175932 Smallest prime p such that there exist exactly n integers b such that 1 < b < p and b^(p-1) == 1 (mod p^2) or, equivalently, Fermat quotient q_p(b) == 0 (mod p).

2, 29, 11, 269, 487, 653, 5107, 103291, 40487, 2544079, 1093, 3511, 1006003

Offset: 0

Max Alekseyev, Oct 24 2010

a(n) is the smallest prime p such that A242830(PrimePi(p)) = n, PrimePi = A000720. - Jianing Song, Jan 27 2019

			a(5) = 653 since 653 is the smallest prime with exactly five bases b = 84, 120, 197, 287, 410.

Richard Fischer, Extreme first bases

Cf. A001220, A130912, A248865, A242830.
Subset of A134307.
Cf. also A255203, A255204, A255205, A255206, A255207, A255208, A255209, A255210.

first_n_entries(n)=v=vector(n); toGo=n; forprime(p=2, , count=sum(b=2, p-1, Mod(b, p^2)^(p-1)==1); if(count<=(n-1)&!v[count+1], v[count+1]=p; toGo--; if(!toGo, return(v)))) \\ Jeppe Stig Nielsen, Jul 31 2015, changed to include a(0) = 2 by Jianing Song, Feb 05 2019

a(0) = 2 prepended by Jianing Song, Jan 27 2019

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.

Original entry on oeis.org

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

Offset: 1

Jeppe Stig Nielsen, Sep 06 2020

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.

			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.

Cf. A134307, A175932, A242830, A248865.

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))

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

cp's OEIS Frontend

A175932 Smallest prime p such that there exist exactly n integers b such that 1 < b < p and b^(p-1) == 1 (mod p^2) or, equivalently, Fermat quotient q_p(b) == 0 (mod p).

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