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%I #11 Sep 07 2018 04:06:26
%S 71,3,11,359,3,11,71,331,359,307,3,11,71,331,359,307,19,3,11,71,331,
%T 487,11,71,331,359,307,487,3,11,71,331,359,307,863,23,13,863,3,11,71,
%U 331,359,23,13,863,3,11,71,331,359,307,19,13,863,467,3,11,71,331
%N Irregular array T(n, k) read by rows, where row n lists the members of n-th Wieferich tuple. Rows are arranged first by size of largest term, then by increasing length of row, then in lexicographic order.
%C Let p_1, p_2, p_3, ..., p_u be a set P of distinct prime numbers and let m_1, m_2, m_3, ..., m_u be a set V of variables. Then P is a Wieferich u-tuple if there exists a mapping from the elements of P to the elements of V such that each of the following congruences is satisfied: m_1^(m_2-1) == 1 (mod (m_2)^2), m_2^(m_3-1) == 1 (mod (m_3)^2), ..., m_u^(m_1-1) == 1 (mod (m_1)^2).
%e Irregular array starts as follows:
%e 71, 3, 11;
%e 359, 3, 11, 71, 331;
%e 359, 307, 3, 11, 71, 331;
%e 359, 307, 19, 3, 11, 71, 331;
%e 487, 11, 71, 331, 359, 307;
%e 487, 3, 11, 71, 331, 359, 307;
%e 863, 23, 13;
%e 863, 3, 11, 71, 331, 359, 23, 13;
%e 863, 3, 11, 71, 331, 359, 307, 19, 13;
%e 863, 467, 3, 11, 71, 331, 359, 23, 13;
%e 863, 3, 11, 71, 331, 359, 307, 487, 23, 13;
%e 863, 467, 3, 11, 71, 331, 359, 307, 19, 13;
%e ...
%e The tuple 359, 3, 11, 71, 331 is a row of the array, because its members satisfy 359^(3-1) == 1 (mod 3^2), 3^(11-1) == 1 (mod 11^2), 11^(71-1) == 1 (mod 71^2), 71^(331-1) == 1 (mod 331^2) and 331^(359-1) == 1 (mod 359^2).
%o (PARI) addtovec(vec) = my(w=[], vmax=0); for(t=1, #vec, if(vecmax(vec[t]) > vmax, vmax=vecmax(vec[t]))); for(k=1, #vec, forprime(q=1, vmax, if(Mod(vec[k][#vec[k]], q^2)^(q-1)==1, w=concat(w, [0]); w[#w]=concat(vec[k], [q])))); w
%o removefromvec(vec) = my(w=[]); for(k=1, #vec, if(vecsort(vec[k])==vecsort(vec[k], , 8), w=concat(w, [0]); w[#w]=vec[k])); w
%o printfromvec(vec) = for(k=1, #vec, if(vec[k][1]==vec[k][#vec[k]], for(t=1, #vec[k]-1, print1(vec[k][t], ", ")); print("")))
%o forprime(p=1, , my(v=[[p]]); while(#v > 0, v=addtovec(v); printfromvec(v); v=removefromvec(v)))
%Y Cf. A271100 (terms of first row of length n), A297846 (distinct terms of column 1 of T), A317919 (number of rows of T with the same largest element), A317920 (length of row n of T).
%K nonn,tabf
%O 1,1
%A _Felix Fröhlich_, Aug 05 2018
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