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A354156 Primes p == 1 (mod 4) which are not Lagrange primes.

%I A354156 #20 May 31 2022 18:23:45
%S A354156 37,61,89,101,109,113,149,157,173,181,193,197,233,269,277,293,317,337,
%T A354156 349,353,373,389,401,421,433,509,557,569,577,593,601,613,641,673,701,
%U A354156 709,757,761,773,797,821,829,877,881,937,941,977,1009,1013,1033,1049,1061
%N A354156 Primes p == 1 (mod 4) which are not Lagrange primes.
%D A354156 J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.5.
%H A354156 Michael S. Branicky, <a href="/A354156/b354156.txt">Table of n, a(n) for n = 1..5383</a>
%o A354156 (Python)
%o A354156 from itertools import islice
%o A354156 from sympy import factorial, nextprime
%o A354156 def agen(): # generator of terms
%o A354156     p = 5
%o A354156     while True:
%o A354156         X = (p-1)//2
%o A354156         Xf = factorial(X)**2
%o A354156         if any(pow(factorial(Y), 2, p)+1 == p for Y in range(X-1, 0, -1)):
%o A354156             yield p
%o A354156         p = nextprime(p)
%o A354156         while p%4 != 1:
%o A354156             p = nextprime(p)
%o A354156 print(list(islice(agen(), 5))) # _Michael S. Branicky_, May 30 2022
%Y A354156 This is the complement of A354155 in A002144.
%K A354156 nonn
%O A354156 1,1
%A A354156 _N. J. A. Sloane_, May 29 2022, based on Section 18.5 of Cosgrave (2022)
%E A354156 a(26) and beyond from _Michael S. Branicky_, May 30 2022