A339539 - OEIS

A339539

T(n, k) is the sum of the residues b^(p-1)-1 (mod p^2) for b=2..k, k > 1, where p = prime(n). Square array, read by antidiagonals, downward.

1, 3, 3, 6, 11, 15, 6, 17, 20, 14, 7, 23, 25, 56, 55, 9, 31, 49, 84, 55, 39, 12, 34, 69, 126, 165, 143, 221, 12, 34, 69, 133, 242, 221, 391, 57, 13, 42, 89, 181, 297, 234, 544, 399, 391, 15, 42, 99, 223, 319, 377, 748, 513, 782, 29, 18, 45, 123, 258, 363, 494

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OFFSET

1,2

COMMENTS

T(n, k) = T(n, k-1) if p is a base-k Wieferich prime.

A necessary condition for the failure of the first case of Fermat's last theorem for a prime p with prime index i is T(i, k) = 0 for k = 2..113 (cf. Suzuki, 1994).

LINKS

Table of n, a(n) for n=1..61.

J. Suzuki, On the generalized Wieferich criteria, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 70, No. 7 (1994), 230-234.

FORMULA

T(n, 2) = A196202(n)-1.

EXAMPLE

The array starts as follows:

1, 3, 6, 6, 7, 9, 12, 12, 13, 15, 18

3, 11, 17, 23, 31, 34, 34, 42, 42, 45, 53

15, 20, 25, 49, 69, 69, 89, 99, 123, 138, 148

14, 56, 84, 126, 133, 181, 223, 258, 265, 279, 300

55, 55, 165, 242, 297, 319, 363, 363, 374, 494, 604