A317721 - OEIS

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.

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

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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

Table of n, a(n) for n=1..60.

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

Adjacent sequences: A317718 A317719 A317720 * A317722 A317723 A317724

KEYWORD

nonn,tabf

AUTHOR

Felix Fröhlich, Aug 05 2018

STATUS

approved