A274916 Triangle T(n, k) read by rows: sum of residues p^(q-1) (mod q^2) and q^(p-1) (mod p^2), where p = prime(n) and q = prime(k) for k = 1, 2, ...., n-1.

7, 17, 13, 18, 47, 44, 59, 5, 94, 38, 41, 112, 25, 133, 242, 223, 172, 226, 74, 188, 204, 61, 344, 250, 249, 128, 344, 317, 395, 399, 339, 306, 262, 347, 398, 412, 31, 440, 355, 835, 757, 737, 300, 713, 772, 190, 535, 301, 808, 137, 1013, 738, 647, 730, 1119

Offset: 1

Author

Felix Fröhlich, Dec 11 2016

Keywords

• nonn
• tabl

Comments

T(n, k) = 2 iff (p, q) is a double Wieferich prime pair.

Examples

			For n = 652 and k = 23: prime(23) = 83 and prime(652) = 4871. 83 and 4871 form a double Wieferich prime pair, so 83^4870 (mod 4871^2) = 1 and 4871^82 (mod 83^2) = 1, hence T(652, 23) = 1+1 = 2.
Triangle starts
    7;
   17,  13;
   18,  47,  44;
   59,   5,  94,  38;
   41, 112,  25, 133, 242;
  223, 172, 226,  74, 188,  204;
   61, 344, 250, 249, 128,  344, 317;
  395, 399, 339, 306, 262,  347, 398, 412;
   31, 440, 355, 835, 757,  737, 300, 713, 772;
  190, 535, 301, 808, 137, 1013, 738, 647, 730, 1119;
		

Links

• Wikipedia, Wieferich pair

Crossrefs

Cf. A124121, A124122, A126432, A266829.

Programs

PARI

t(n, k) = lift(Mod(prime(n), prime(k)^2)^(prime(k)-1)) + lift(Mod(prime(k), prime(n)^2)^(prime(n)-1))
trianglerows(n) = for(x=2, n+1, for(y=1, x-1, print1(t(x, y), ", ")); print(""))
trianglerows(6) \\ print upper 6 rows of triangle