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A309284
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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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1
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OFFSET
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1,1
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COMMENTS
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a(n) is an Euler pseudoprime to base 2, so it is also a Fermat pseudoprime to base 2.
This sequence is analogous to the sequence A000229 of primes.
Conjecture: the smallest quadratic non-residue modulo a(n) is prime(n), i.e., A020649(a(n)) = prime(n).
a(10) <= 41154189126635405260441. - Daniel Suteu, Jul 22 2019
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LINKS
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FORMULA
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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.
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PROG
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(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)));
a(n) = for(k=9, oo, if(isok(n, k), return(k))); \\ Daniel Suteu, Jul 22 2019
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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