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

Offset: 1

Author

Felix Fröhlich, Aug 05 2018

Keywords

• nonn
• tabf

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).

Examples

			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).
		

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).

Programs

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)))