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A317721 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. 11
71, 3, 11, 359, 3, 11, 71, 331, 359, 307, 3, 11, 71, 331, 359, 307, 19, 3, 11, 71, 331, 487, 11, 71, 331, 359, 307, 487, 3, 11, 71, 331, 359, 307, 863, 23, 13, 863, 3, 11, 71, 331, 359, 23, 13, 863, 3, 11, 71, 331, 359, 307, 19, 13, 863, 467, 3, 11, 71, 331 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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).
LINKS
EXAMPLE
Irregular array starts as follows:
71, 3, 11;
359, 3, 11, 71, 331;
359, 307, 3, 11, 71, 331;
359, 307, 19, 3, 11, 71, 331;
487, 11, 71, 331, 359, 307;
487, 3, 11, 71, 331, 359, 307;
863, 23, 13;
863, 3, 11, 71, 331, 359, 23, 13;
863, 3, 11, 71, 331, 359, 307, 19, 13;
863, 467, 3, 11, 71, 331, 359, 23, 13;
863, 3, 11, 71, 331, 359, 307, 487, 23, 13;
863, 467, 3, 11, 71, 331, 359, 307, 19, 13;
...
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).
PROG
(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
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
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("")))
forprime(p=1, , my(v=[[p]]); while(#v > 0, v=addtovec(v); printfromvec(v); v=removefromvec(v)))
CROSSREFS
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).
Sequence in context: A126649 A266548 A051324 * A350007 A128856 A033391
KEYWORD
nonn,tabf
AUTHOR
Felix Fröhlich, Aug 05 2018
STATUS
approved

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Last modified March 29 10:59 EDT 2024. Contains 371277 sequences. (Running on oeis4.)