A307639 Irregular array T(n, k) read by rows, where row n lists the members of n-th Fermat pseudoprime tuple. Rows are arranged first by size of largest term, then by increasing length of row, then in lexicographic order.

9, 8, 15, 14, 21, 20, 21, 10, 9, 8, 25, 24, 27, 26, 28, 9, 28, 27, 26, 9, 28, 27, 26, 25, 28, 27, 26, 25, 8, 9, 28, 27, 26, 25, 8, 21, 10, 9, 33, 32, 33, 8, 21, 10, 33, 32, 25, 8, 21, 10, 33, 32, 25, 28, 9, 8, 21, 10, 33, 32, 25, 28, 27, 26, 9, 8, 21, 10

Offset: 1

Author

Felix Fröhlich, Apr 19 2019

Keywords

• nonn
• tabf

Comments

Let c_1, c_2, c_3, ..., c_u be a set C of distinct composites and let m_1, m_2, m_3, ..., m_u be a set V of variables. Then C is a Fermat pseudoprime u-tuple if there exists a mapping from the elements of C to the elements of V such that each of the following congruences is satisfied: m_1^(m_2-1) == 1 (mod m_2), m_2^(m_3-1) == 1 (mod m_3), ..., m_u^(m_1-1) == 1 (mod m_1).

Examples

			Irregular array starts as follows:
   9,  8
  15, 14
  21, 20
  21, 10, 9, 8
  25, 24
  27, 26
  28,  9
  28, 27, 26,  9
  28, 27, 26, 25
  28, 27, 26, 25, 8,  9
  28, 27, 26, 25, 8, 21, 10, 9
  33, 32
  33,  8, 21, 10
  33, 32, 25,  8, 21, 10
  33, 32, 25, 28,  9,  8, 21, 10
  33, 32, 25, 28, 27, 26,  9,  8, 21, 10
  35, 34
  35, 34, 33, 32, 25,  6
  35,  9, 28, 27, 26, 25,  6
  35, 34, 21, 10, 33, 32, 25,  6
  35,  9,  8, 21, 10, 33, 32, 25,  6
  35, 34, 21, 10,  9, 28, 27, 26, 25,  6
  35, 34, 33,  8,  9, 28, 27, 26, 25,  6
  35, 34, 21, 10, 33,  8,  9, 28, 27, 26, 25,  6
  35, 34, 33,  8, 21, 10,  9, 28, 27, 26, 25,  6
  39, 38
The composites 21, 10, 9 and 8 satisfy the congruences 21^(10-1) == 1 (mod 10), 10^(9-1) == 1 (mod 9), 9^(8-1) == 1 (mod 8) and 8^(21-1) == 1 (mod 21), so 21, 10, 9, 8 is a row of the array.
		

Crossrefs

Cf. A317721.

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, forcomposite(c=1, vmax, if(Mod(vec[k][#vec[k]], c)^(c-1)==1, w=concat(w, [0]); w[#w]=concat(vec[k], [c])))); 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("")))
forcomposite(c=1, 40, my(v=[[c]]); while(#v > 0, v=addtovec(v); printfromvec(v); v=removefromvec(v)))