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%I #12 Jul 01 2026 06:46:15
%S 11,17,307,66431,7036306088681
%N Prime numbers p such that p^2 divides 40^(p-1) - 1.
%C a(5) = 7036306088681 is from Fischer's tables. No more terms up to 2*10^14.
%H Richard Fischer, <a href="http://www.fermatquotient.com/FermatQuotienten/FermQ_Sort.txt">Fermatquotienten von 2 bis 1052</a>, Dec 19 2019.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Wieferich_prime">Wieferich prime</a>
%o (PARI) forprime(p=2, 1e8, if(Mod(40, p^2)^(p-1)==1, print1(p", ")))
%Y Wieferich primes to base b: A001220 (b=2), A014127 (b=3), A123692 (b=5), A212583 (b=6), A123693 (b=7), A045616 (b=10), A111027 (b=12), A128667 (b=13), A234810 (b=14), A242741 (b=15), A128668 (b=17), A244260 (b=18), A090968 (b=19), A242982 (b=20), A298951 (b=22), A128669 (b=23), A306255 (b=26), A306256 (b=30), A331424 (b=31), A396350 (b=33), A397319 (b=35), A331426 (b=37), this sequence (b=40), A331427 (b=41).
%Y Cf. A039951.
%K nonn,hard,more,changed
%O 1,1
%A _Daniel Okwor_, Jun 20 2026