A253684
Primes q with A253683(n) > q > A253685(n) such that (A253683(n), q, A253685(n)) forms a Wieferich triple.
Original entry on oeis.org
11, 23, 5, 1667, 73, 821, 18043, 2393, 20771, 2251, 1006003
Offset: 1
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forprime(p=1, , forprime(q=1, p, forprime(r=1, q, if((Mod(p, q^2)^(q-1)==1 && Mod(q, r^2)^(r-1)==1 && Mod(r, p^2)^(p-1)==1) || (Mod(p, r^2)^(r-1)==1 && Mod(r, q^2)^(q-1)==1 && Mod(q, p^2)^(p-1)==1), print1(q, ", ")))))
A253685
Primes r with A253683(n) > A253684(n) > r such that (A253683(n), A253684(n), r) is a Wieferich triple.
Original entry on oeis.org
3, 13, 2, 1657, 2, 83, 5, 431, 5, 199, 3
Offset: 1
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forprime(p=1, , forprime(q=1, p, forprime(r=1, q, if((Mod(p, q^2)^(q-1)==1 && Mod(q, r^2)^(r-1)==1 && Mod(r, p^2)^(p-1)==1) || (Mod(p, r^2)^(r-1)==1 && Mod(r, q^2)^(q-1)==1 && Mod(q, p^2)^(p-1)==1), print1(r, ", ")))))
A297846
Primes p such that p is the largest member of a Wieferich tuple.
Original entry on oeis.org
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
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 column 1 of
A271100.
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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, ", ")))
A271100
Triangular array read by rows: T(n, k) = k-th largest member of lexicographically earliest Wieferich n-tuple that contains no duplicate members, read by rows, or T(n, k) = 0 if no Wieferich n-tuple exists.
Original entry on oeis.org
0, 1093, 2, 71, 11, 3, 3511, 19, 13, 2, 359, 331, 71, 11, 3, 359, 331, 307, 71, 11, 3, 359, 331, 307, 71, 19, 11, 3, 863, 359, 331, 71, 23, 13, 11, 3, 863, 359, 331, 307, 71, 19, 13, 11, 3, 863, 467, 359, 331, 307, 71, 19, 13, 11, 3
Offset: 1
For n = 1: There is no Wieferich singleton (1-tuple), because no prime p satisfies the congruence p^(p-1) == 1 (mod p^2), so T(1, 1) = 0.
For n = 4: The primes 3511, 19, 13, 2 satisfy the congruences 3511^(19-1) == 1 (mod 19^2), 19^(13-1) == 1 (mod 13^2), 13^(2-1) == 1 (mod 2^2) and 2^(3511-1) == 1 (mod 3511^2) and thus form a "Wieferich quadruple". Since this is the lexicographically earliest such set of primes, T(4, 1..4) = 3511, 19, 13, 2.
Triangle starts:
n=1: 0;
n=2: 1093, 2;
n=3: 71, 11, 3;
n=4: 3511, 19, 13, 2;
n=5: 359, 331, 71, 11, 3;
n=6: 359, 331, 307, 71, 11, 3;
n=7: 359, 331, 307, 71, 19, 11, 3;
n=8: 863, 359, 331, 71, 23, 13, 11, 3;
n=9: 863, 359, 331, 307, 71, 19, 13, 11, 3;
n=10: 863, 467, 359, 331, 307, 71, 19, 13, 11, 3;
....
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ulimupto(u,{llim=2}) = {my(l=List());
forprime(i=nextprime(llim+1),u,if(Mod(llim,i^2)^(i-1)==1,listput(l,i)));l} \\ David A. Corneth, May 14 2016
\\tests if a tuple is a valid Wieferich n-tuple.
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istuple(v) = {if(#Set(v)==#v,return(0));for(j=0,(#v-1)!-1, w=vector(#v,k,v[numtoperm(#v,j)[k]]); if(sum(i=2,#w,Mod(w[i-1],w[i]^2)^(w[i]-1)==1)+(Mod(w[1],w[#w])^(w[#w]-1)==1)==#w,return(1)));0} \\ David A. Corneth, May 15 2016
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wief = DiGraph([prime_range(3600), lambda p, q: power_mod(p, q-1, q^2)==1])
sc = [[0]] + [sorted(c[1:], reverse=1) for c in wief.all_simple_cycles()]
tbl = [sorted(filter(lambda c: len(c)==n, sc))[0] for n in range(1, len(sc[-1]))]
for t in tbl: print('n=%d:'% len(t), ', '.join("%s"%i for i in t)) # Bruce Leenstra, May 18 2016
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