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

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
   39, 143,  221,  234,  377,  494,  611,  650,  702,  832,  845
  221, 391,  544,  748,  850,  901,  986, 1037, 1173, 1326, 1360
   57, 399,  513,  741,  779, 1026, 1197, 1520, 1805, 1843, 1938
  391, 782, 1035, 1357, 1610, 1886, 2001, 2254, 2438, 2599, 2714
   29, 464,  522,  870, 1334, 2146, 2233, 2262, 2639, 3306, 3799
  186, 713, 1085, 1364, 2077, 2883, 3441, 3534, 3999, 4123, 5022
   37, 703,  777, 1776, 2479, 3589, 3700, 5032, 6068, 6512, 7252

Prog

(PARI) t(n, k) = my(p=prime(n)); sum(b=2, k, lift(Mod(b, p^2)^(p-1)-1))
array(rows, cols) = for(x=1, rows, for(y=2, cols+1, print1(t(x, y), ", ")); print(""))
array(11, 12) \\ Print initial 11 rows and 12 columns of array

Crossrefs

Cf. A196202.

Sequence in context: A298954 A169944 A110952 * A025250 A326498 A367644

Adjacent sequences: A339536 A339537 A339538 * A339540 A339541 A339542

Keyword

nonn,tabl

Author

Felix Fröhlich, Dec 08 2020

Status

approved