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A275118 Split primes p such that prime P lying above p is a Wieferich place of K (with discriminant D_K), for some imaginary quadratic field K of class number 1.

%I A275118 #17 May 05 2024 08:57:04
%S A275118 5,11,13,181,1523,1741,2521,19531,24421,29789,76543,108529,489061,
%T A275118 880301,1769069,6811741
%N A275118 Split primes p such that prime P lying above p is a Wieferich place of K (with discriminant D_K), for some imaginary quadratic field K of class number 1.
%H A275118 D. S. Dummit, D. Ford, H. Kisilevsky, and J. W. Sands, <a href="https://doi.org/10.1016/S0022-314X(05)80027-7">Computation of Iwasawa Lambda invariants for imaginary quadratic fields</a>, Journal of Number Theory, Vol. 37, No. 1 (1991), 100-121.
%H A275118 Á. Lozano-Robledo, <a href="http://dx.doi.org/10.1016/j.jnt.2009.10.013">Bernoulli-Hurwitz numbers, Wieferich primes and Galois representations</a>, Journal of Number Theory, Vol. 130, No. 3 (2010), 539-558. See table 2 on page 555.
%o A275118 (Sage)
%o A275118 def is_A275118(k):
%o A275118     if not Integer(k).is_prime(): return False
%o A275118     for D in [1, 2, 3, 7, 11, 19, 43, 67, 163]:
%o A275118         fct = QuadraticField(-D).ideal(k).factor()
%o A275118         if len(fct)==2:
%o A275118             pi = fct[1][0].gens_reduced()[0]
%o A275118             if (pi^(k-1) - 1).valuation(fct[0][0]) > 1: return True
%o A275118     return False
%o A275118 print([k for k in range(10^7) if is_A275118(k)])  # _Robin Visser_, Apr 26 2024
%Y A275118 Cf. A239902.
%K A275118 nonn,more
%O A275118 1,1
%A A275118 _Felix Fröhlich_, Jul 18 2016
%E A275118 a(11)-a(16) from _Robin Visser_, Apr 26 2024