%I #37 Oct 12 2021 15:41:51
%S 1,1,1,3,5,3,-1,3,-8,-3,-6,13,-2,-4,16,-25,10,-13,7,-16,-15,-30,21,5,
%T 37,-4,22,24,26,-53,13,64,58,-22,-29,60,44,-3,44,-43,-5,-50,94,31,-56,
%U 5,-99,3,-73,18,29,5,-59,-1,2
%N Let p be the n-th prime, then a(n) = A/p where A is the smallest number (in absolute value) such that F_{p-(p/5)} == A (mod p^2) with F_n = A000045(n) and (p/5) the Legendre symbol.
%C a(n) is the smallest A such that p is a near-Wall-Sun-Sun prime. A gives the value of F_p-(p/5) modulo p^2 and a value of 0 would indicate a Wall-Sun-Sun prime. A244801 is similar but always gives the positive A, while this sequence gives A with the smallest absolute value.
%C a(1), with p=2, is technically ambiguous between 1 and -1, so a(1)=1 is by convention. Clearly this cannot happen for n>1 (where p^2 is odd). - _Jeppe Stig Nielsen_, Sep 09 2021
%H Michael De Vlieger, <a href="/A241014/b241014.txt">Table of n, a(n) for n = 1..10000</a>
%H R. J. McIntosh and E. L. Roettger, <a href="http://dx.doi.org/10.1090/S0025-5718-07-01955-2">A search for Fibonacci-Wieferich and Wolstenholme primes</a>, Math. Comp., 76 (2007), 2087-2094.
%t Array[(#3 - #2 Boole[#3 > #2/2])/#1 & @@ {#, #^2, Mod[Fibonacci[# - KroneckerSymbol[#, 5]], #^2]} &@ Prime[#] &, 55] (* _Michael De Vlieger_, Sep 08 2021 *)
%o (PARI) forprime(p=2, 1e2, a=fibonacci(p-kronecker(p, 5))%p^2; if(a>p^2/2, a-=p^2); a=a/p; print1(a, ", "))
%o (PARI) a(n)=my(p=prime(n)); centerlift(((Mod([1, 1; 1, 0], p^2))^(p-kronecker(p,5))))[1, 2]/p \\ _Charles R Greathouse IV_, Aug 21 2014
%Y Cf. A113650, A195988, A244801, A339855, A347565.
%K sign
%O 1,4
%A _Felix Fröhlich_, Aug 13 2014