A271100 -id:A271100 - cp's OEIS frontend

A297846 Primes p such that p is the largest member of a Wieferich tuple.

71, 359, 487, 863, 1069, 1093, 1483, 1549, 2281, 3511, 4871, 6451, 6733, 7393, 12049, 13691, 14107, 14149, 15377, 17401, 18787, 20771, 29573, 32933, 35747, 39233, 44483, 46021, 48947, 49559, 54787, 54979, 59197, 60493, 69401, 69653, 77263, 77867, 105323, 122327

Offset: 1

Felix Fröhlich, Jan 07 2018

nonn

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).

			The primes 31, 79, 251, 263, 421 and 1483 satisfy 31^(79-1) == 1 (mod 79^2), 79^(263-1) == 1 (mod 263^2), 263^(251-1) == 1 (mod 251^2), 251^(421-1) == 1 (mod 421^2), 421^(1483-1) == 1 (mod 1483^2) and 1483^(31-1) == 1 (mod 31^2), so those primes form a Wieferich tuple. Since 1483 is the largest member of the tuple, 1483 is a term of the sequence.

Supersequence of A253683, A266829 and A289899.

Supersequence of column 1 of A271100.

findwiefs(vec, lim) = my(v=[]); for(k=1, #vec, forprime(p=1, lim, if(Mod(vec[k], p^2)^(p-1)==1, v=concat(v, [p])))); vecsort(v, , 8)
newprimes(v, w) = setminus(w, v)
is(n) = my(v=findwiefs([n], n), w=[], np=[]); while(1, w=findwiefs(v, n); if(newprimes(v, w)==[], return(0), if(setsearch(vecsort(newprimes(v, w)), n) > 0, return(1))); v=concat(v, newprimes(v, w)); v=vecsort(v, , 8))
forprime(p=1, , if(is(p), print1(p, ", ")))

More terms from Felix Fröhlich, Jan 22 2018

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

Felix Fröhlich, Aug 05 2018

			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).

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)))

A289899 Primes that are the largest member of a Wieferich cycle.

Original entry on oeis.org

71, 1093, 4871

Offset: 1

Felix Fröhlich, Jul 14 2017

A Wieferich cycle is a repeating cycle in the trajectory of p under successive applications of the map p -> A039951(p), i.e., a part of a row of A288097 repeating indefinitely.
The above cycles could more precisely be called "order-1 Wieferich cycles". A cycle in a row of A281002 could be called an "order-2 Wieferich cycle".
The cycles corresponding to a(1)-a(3) are {3, 11, 71}, {2, 1093} and {83, 4871}, respectively.
The order of the cycle is not to be confused with its length. The order-1 cycle {3, 11, 71} is a cycle of length 3, while the order-1 cycles {2, 1093} and {83, 4871} are cycles of length 2.
Wieferich cycles are special cases of Wieferich tuples (cf. A271100).
a(4) > 20033669 if it exists.

			71 is a term, since A039951(71) = 3, A039951(3) = 11 and A039951(11) = 71, so {3, 11, 71} is a Wieferich cycle of length 3 and 71 is the largest member of that cycle.

Cf. A039951, A252801, A252802, A252812, A268479, A269111, A271100, A281002, A288097.

leastwieferich(base, bound) = forprime(p=1, bound, if(Mod(base, p^2)^(p-1)==1, return(p))); 0
is(n) = my(v=[leastwieferich(n, n)]); while(1, if(v[#v]==0, return(0), v=concat(v, leastwieferich(v[#v], n))); my(x=#v-1); while(x > 1, if(v[#v]==v[x], if(n==vecmax(v), return(1), return(0))); x--))
forprime(p=1, , if(is(p), print1(p, ", ")))

cp's OEIS Frontend

A297846 Primes p such that p is the largest member of a Wieferich tuple.

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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.

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PARI

A289899 Primes that are the largest member of a Wieferich cycle.

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Author

Keywords

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Examples

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Programs

PARI