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

Author

Felix Fröhlich, Jan 07 2018

Keywords

• nonn

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

			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.
		

Crossrefs

Supersequence of A253683, A266829 and A289899.

Supersequence of column 1 of A271100.

Programs

PARI

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, ", ")))

Extensions

More terms from Felix Fröhlich, Jan 22 2018