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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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).

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;

....

LINKS

Bruce Leenstra, Table of n, a(n) for n = 1..300

Wikipedia, Wieferich pair

EXAMPLE

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.

For finding candidate values for m_1 given some m_u, one checks primes higher than m_u for primes satisfying m_u^(m_1-1) == 1 (mod (m_1)^2). For example, to see what we could get if m_u = 2, we check up to say m_1 = 1,000,000 to get candidates for m_1. This would give m_1 in {1093, 3511}. - David A. Corneth, May 14 2016

PROG

\\finds candidate values for the highest value of a tuple up to some value u.

(PARI) 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.

(PARI) 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

(Sage) 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

CROSSREFS

Cf. A124121, A124122, A253683, A253684, A253685, A266829.

Sequence in context: A091673 A288097 A281001 * A258368 A174422 A255838

Adjacent sequences:  A271097 A271098 A271099 * A271101 A271102 A271103

KEYWORD

nonn,tabl

AUTHOR

Felix Fröhlich, Mar 30 2016

EXTENSIONS

a(11)-a(15) from Felix Fröhlich, Apr 26 2016

More terms from Bruce Leenstra, May 18 2016

STATUS

approved

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Last modified October 16 07:30 EDT 2019. Contains 328051 sequences. (Running on oeis4.)