A317919
Number of Wieferich tuples with A297846(n) as largest member, i.e., number of rows of the array in A317721 where A297846(n) is the largest element of that row.
Original entry on oeis.org
1, 3, 2, 7, 1, 2, 1, 6, 1, 317, 1, 1, 230, 580
Offset: 1
For n = 2: The second prime that occurs in column 1 of the array in A317721 is 359. 359 occurs as the largest element in 3 rows of the array, so a(2) = 3.
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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
forprime(p=1, , my(v=[[p]], i=0); while(#v > 0, v=addtovec(v); for(k=1, #v, if(v[k][1]==v[k][#v[k]], i++)); v=removefromvec(v)); if(i > 0, print1(i, ", ")))
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.
Original entry on oeis.org
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
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).
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).
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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)))
A317920
Length of row n of A317721, i.e., number of elements in n-th Wieferich tuple when ordering the tuples as in A317721.
Original entry on oeis.org
3, 5, 6, 7, 6, 7, 3, 8, 9, 9, 10, 10, 11, 9, 2, 3, 6, 9, 10, 11, 12, 13, 14, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12
Offset: 1
For n = 1: Row 1 of A317721 has 3 elements, i.e., the first Wieferich tuple listed in A317721 is a "Wieferich triple", so a(1) = 3.
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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]], print1(#vec[k]-1, ", ")))
forprime(p=1, , my(v=[[p]]); while(#v > 0, v=addtovec(v); printfromvec(v); v=removefromvec(v)))
Original entry on oeis.org
71, 359, 359, 359, 487, 487, 863, 863, 863, 863, 863, 863, 863, 1069, 1093, 1093, 1483, 1549, 1549, 1549, 1549, 1549, 1549, 2281, 3511, 3511, 3511, 3511, 3511, 3511, 3511, 3511, 3511, 3511, 3511, 3511, 3511, 3511, 3511, 3511, 3511, 3511, 3511, 3511, 3511, 3511
Offset: 1
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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]], print1(vec[k][1], ", ")))
forprime(p=1, , my(v=[[p]]); while(#v > 0, v=addtovec(v); printfromvec(v); v=removefromvec(v)))
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