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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. 1

%I

%S 7,17,13,18,47,44,59,5,94,38,41,112,25,133,242,223,172,226,74,188,204,

%T 61,344,250,249,128,344,317,395,399,339,306,262,347,398,412,31,440,

%U 355,835,757,737,300,713,772,190,535,301,808,137,1013,738,647,730,1119

%N 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.

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

%C Triangle starts

%C 7;

%C 17, 13;

%C 18, 47, 44;

%C 59, 5, 94, 38;

%C 41, 112, 25, 133, 242;

%C 223, 172, 226, 74, 188, 204;

%C 61, 344, 250, 249, 128, 344, 317;

%C 395, 399, 339, 306, 262, 347, 398, 412;

%C 31, 440, 355, 835, 757, 737, 300, 713, 772;

%C 190, 535, 301, 808, 137, 1013, 738, 647, 730, 1119;

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Wieferich_pair">Wieferich pair</a>

%e 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.

%o (PARI) t(n, k) = lift(Mod(prime(n), prime(k)^2)^(prime(k)-1)) + lift(Mod(prime(k), prime(n)^2)^(prime(n)-1))

%o trianglerows(n) = for(x=2, n+1, for(y=1, x-1, print1(t(x, y), ", ")); print(""))

%o trianglerows(6) \\ print upper 6 rows of triangle

%Y Cf. A124121, A124122, A126432, A266829.

%K nonn,tabl

%O 1,1

%A _Felix Fröhlich_, Dec 11 2016

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Last modified June 5 08:21 EDT 2020. Contains 334829 sequences. (Running on oeis4.)