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A339639 a(n) is the sum of the Wieferich and Wall-Sun-Sun residues of prime(n).

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%I A339639 #7 Dec 21 2020 07:17:34
%S A339639 3,6,20,35,110,78,493,114,736,783,961,518,2542,2752,2820,3392,1062,
%T A339639 5124,1139,4047,8322,5372,5727,979,9118,19089,8343,3959,10137,16159,
%U A339639 3937,10611,15207,20433,32184,17516,19782,37001,15197,23009,40096,50499,27504,26055
%N A339639 a(n) is the sum of the Wieferich and Wall-Sun-Sun residues of prime(n).
%C A339639 If a(n) = 0 then prime(n) is both a Wieferich prime (A001220) and a Wall-Sun-Sun (Fibonacci-Wieferich) prime.
%C A339639 If the first case of Fermat's last theorem fails for a prime p, that prime is both a Wieferich prime (cf. Wieferich, 1909) and a Wall-Sun-Sun prime (cf. Sun, 1992).
%H A339639 Zhi-Wei Sun, <a href="https://doi.org/10.4064/aa-60-4-371-388">Fibonacci numbers and Fermat's last theorem</a>, Acta Arithemtica, Vol. 60, No. 4 (1992), 371-388.
%H A339639 A. Wieferich, <a href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002166968">Zum letzten Fermat'schen Theorem</a>, Journal für die reine und angewandte Mathematik, 136 (1909), 293-302, DOI:<a href="https://doi.org/10.1515/crll.1909.136.293">10.1515/crll.1909.136.293</a>.
%F A339639 a(n) = A196202(n) + A113650(n) - 1.
%o A339639 (PARI) a(n) = my(p=prime(n)); lift(Mod([1, 1; 1, 0]^(p-kronecker(p, 5)), p^2)[1, 2]) + lift(Mod(2, p^2)^(p-1)) - 1
%Y A339639 Cf. A001220, A113650, A196202.
%K A339639 nonn
%O A339639 1,1
%A A339639 _Felix Fröhlich_, Dec 11 2020