cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A309284 a(n) is the smallest odd composite k such that prime(n)^((k-1)/2) == -1 (mod k) and b^((k-1)/2) == 1 (mod k) for every natural b < prime(n).

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%I A309284 #30 Jul 18 2022 22:48:42
%S A309284 3277,5173601,2329584217,188985961,5113747913401,30990302851201,
%T A309284 2528509579568281,5189206896360728641,12155831039329417441
%N A309284 a(n) is the smallest odd composite k such that prime(n)^((k-1)/2) == -1 (mod k) and b^((k-1)/2) == 1 (mod k) for every natural b < prime(n).
%C A309284 a(n) is an Euler pseudoprime to base 2, so it is also a Fermat pseudoprime to base 2.
%C A309284 This sequence is analogous to the sequence A000229 of primes.
%C A309284 Conjecture: the smallest quadratic non-residue modulo a(n) is prime(n), i.e., A020649(a(n)) = prime(n).
%C A309284 a(10) <= 41154189126635405260441. - _Daniel Suteu_, Jul 22 2019
%F A309284 According to the data, b^((a(n)-1)/2) == (b / a(n)) (mod a(n)) for every natural b <= prime(n), where (x / y) is the Jacobi symbol.
%o A309284 (PARI) isok(n,k) = (k%2==1) && !isprime(k) && Mod(prime(n), k)^((k-1)/2) == Mod(-1, k) && !for(b=2, prime(n)-1, if(Mod(b, k)^((k-1)/2) != Mod(1, k), return(0)));
%o A309284 a(n) = for(k=9, oo, if(isok(n, k), return(k))); \\ _Daniel Suteu_, Jul 22 2019
%Y A309284 Cf. A000229, A001567, A006970, A307767, A309285.
%K A309284 nonn,hard,more
%O A309284 1,1
%A A309284 _Thomas Ordowski_, Jul 21 2019
%E A309284 a(5)-a(9) from _Amiram Eldar_, Jul 21 2019