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Primes p where A242830(i) reaches record values, where i is the index of p in A000040.

The corresponding number of bases are 0, 2, 3, 4, 5, 10, 11, 12. - Jianing Song, Feb 07 2019

From Jeppe Stig Nielsen, Sep 06 2020: (Start)

Note that for a(6) and a(7), all the b values (bases) that are counted are powers of 2; and for a(8) all are powers of 3.

See A334048 for a version where bases b that are powers are not allowed.

One candidate for a(9) is 1645333507; it has 14 bases, the first 13 of which are powers of 5. However, excluding all numbers under 1645333507 as candidates for a(9) may be difficult to do.

(End)

From Robert G. Wilson v, Mar 11 2015: (Start)

n b p

1: 5 {2}

2: 17 {2, 3}

3: 19 {3, 7, 13}

4: 233 {2, 3, 11, 157}

5: 293 {2, 5, 7, 19, 83}

6: 977 {2, 11, 17, 109, 239, 401}

7: 1451 {5, 7, 11, 13, 83, 173, 1259}

8: 1693 {2, 3, 5, 11, 31, 37, 61, 109}

9: 33301 {2, 3, 5, 7, 43, 293, 317, 383, 1627}

10: 308093 {2, 5, 7, 11, 31, 47, 89, 167, 523, 619}

... (End)

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.