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A350398
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Numbers k such that for all pairs of primes p,q with p+q = 2*k, p*q mod 2*k is prime.
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1
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1, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 21, 24, 30, 31, 34, 36, 42, 45, 49, 63
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OFFSET
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1,2
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COMMENTS
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a(25) > 10^6 if it exists.
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LINKS
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EXAMPLE
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a(5) = 7 is a term because 2*7 = 14 = 3+11 = 7+7, with 3*11 == 5 (mod 14) and 7*7 == 7 (mod 14), and both 5 and 7 are prime.
5 is not a term because 2*5 = 10 = 3+7 = 5+5, but 3*7 == 1 (mod 10) and 1 is not prime.
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MAPLE
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filter:= proc(k) local p;
p:= 1;
while p <= k do
p:= nextprime(p);
if isprime(2*k-p) and not isprime(-p^2 mod 2*k) then return false fi
od;
true
end proc:
select(filter, [$1..1000]);
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MATHEMATICA
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q[k_] := AllTrue[Select[Range[2, 2*k], PrimeQ], ! PrimeQ[2*k - #] || PrimeQ[Mod[#*(2*k - #), 2*k]] &]; Select[Range[100], q] (* Amiram Eldar, Dec 28 2021 *)
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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