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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 downward antidiagonals.
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%I #13 Oct 20 2024 11:40:40

%S 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,

%T 165,143,221,12,34,69,133,242,221,391,57,13,42,89,181,297,234,544,399,

%U 391,15,42,99,223,319,377,748,513,782,29,18,45,123,258,363,494

%N 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 downward antidiagonals.

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

%C 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).

%H J. Suzuki, <a href="https://doi.org/10.3792/pjaa.70.230">On the generalized Wieferich criteria</a>, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 70, No. 7 (1994), 230-234.

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

%e The array starts as follows:

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

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

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

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

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

%e 39, 143, 221, 234, 377, 494, 611, 650, 702, 832, 845

%e 221, 391, 544, 748, 850, 901, 986, 1037, 1173, 1326, 1360

%e 57, 399, 513, 741, 779, 1026, 1197, 1520, 1805, 1843, 1938

%e 391, 782, 1035, 1357, 1610, 1886, 2001, 2254, 2438, 2599, 2714

%e 29, 464, 522, 870, 1334, 2146, 2233, 2262, 2639, 3306, 3799

%e 186, 713, 1085, 1364, 2077, 2883, 3441, 3534, 3999, 4123, 5022

%e 37, 703, 777, 1776, 2479, 3589, 3700, 5032, 6068, 6512, 7252

%o (PARI) t(n, k) = my(p=prime(n)); sum(b=2, k, lift(Mod(b, p^2)^(p-1)-1))

%o array(rows, cols) = for(x=1, rows, for(y=2, cols+1, print1(t(x, y), ", ")); print(""))

%o array(11, 12) \\ Print initial 11 rows and 12 columns of array

%Y Cf. A196202.

%K nonn,tabl

%O 1,2

%A _Felix Fröhlich_, Dec 08 2020