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A350399
a(n) is the number of prime pairs (p,q) with p <= q, p+q = 2*n, and p*q mod (2*n) prime.
3
0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 1, 3, 1, 2, 4, 1, 2, 5, 2, 1, 4, 2, 2, 6, 3, 3, 4, 2, 4, 6, 2, 3, 5, 3, 4, 8, 3, 1, 9, 2, 3, 6, 3, 4, 5, 2, 4, 6, 4, 4, 8, 5, 2, 7, 3, 3, 10, 1, 2, 6, 2, 2, 6, 5, 4, 5, 3, 3, 11, 1, 4, 8, 4, 4, 7, 2, 5, 8, 4, 2, 7, 4, 1, 12, 4, 2, 9, 3, 4, 7, 2, 5
OFFSET
1,7
COMMENTS
Conjecture: a(n) > 0 for n >= 3.
a(n) <= A002375(n) with equality for n in A350398.
LINKS
EXAMPLE
a(7) = 2 because there are 2 such pairs, namely 14 = 3+13 = 7+7 with 3*13 == 5 (mod 14) and 7*7 == 7 (mod 14).
MAPLE
f:= proc(k) local P, i;
P:= select(t -> isprime(t) and isprime(2*k-t) and isprime(-t^2 mod (2*k)), [2, seq(i, i=3..k, 2)]);
nops(P);
end proc:
map(f, [$1..100]);
MATHEMATICA
a[n_] := Count[Select[Range[2, 2*n], PrimeQ], _?(# >= n && PrimeQ[2*n - #] && PrimeQ[Mod[#*(2*n - #), 2*n]] &)]; Array[a, 100] (* Amiram Eldar, Dec 28 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
J. M. Bergot and Robert Israel, Dec 28 2021
STATUS
approved