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A258367
a(n) is the smallest A (in absolute value) such that for p = prime(n), 2^{(p-1)/2} == +-1 + A*p (mod p^2), i.e., such that p is a near-Wieferich prime.
16
1, 1, 1, 3, 5, 2, 8, 3, 14, 3, 18, 9, 9, 22, 18, 4, 18, 5, 1, 28, 30, 24, 3, 20, 46, 22, 47, 21, 15, 9, 57, 42, 15, 48, 28, 41, 48, 60, 85, 25, 74, 25, 52, 11, 32, 51, 17, 13, 34, 113, 13, 71, 2, 16, 64, 130, 81, 35, 37, 29, 39, 147, 68, 60, 71, 96, 92, 99, 12
OFFSET
2,4
COMMENTS
p is in A001220 iff a(n) = 0. This is the case iff A014664(n) = A243905(n), which happens for n = 183 and n = 490.
Is a(n) = 0 for any other n, and, if yes, are there infinitely many such n?
LINKS
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation, 66 (1997), 433-449.
FORMULA
a(n) = min(b(n) mod p, -b(n) mod p) where p = prime(n) and b(n) = Sum_{i=1..ceiling((p-1)/4)} (2i-1)^(p-2). - Daniel Chen, Sep 01 2022
PROG
(PARI) a(n, p=prime(n))=abs(centerlift(Mod(2, p^2)^((p-1)/2))\/p)
apply(p->a(0, p), primes(100)[2..100]) \\ Charles R Greathouse IV, Jun 15 2015
KEYWORD
nonn
AUTHOR
Felix Fröhlich, May 28 2015
STATUS
approved