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 A350544 a(n) is the least prime p such that there exists a prime q with p^2 + n = (n+1)*q^2, or 0 if there is no such p. 2
 7, 5, 0, 11, 7, 13, 5, 0, 41, 23, 17, 10496997797584752004430879, 41, 11, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS a(16) > 10^1000 if it is not 0. If it is not 0, then a(16) = A199773(k) where k is the smallest index such that both p = A199773(k) and q = A199772(k) are prime. If such an index exists, a(16) > 10^10000. - Jon E. Schoenfield, Jan 11 2022 LINKS Table of n, a(n) for n=1..15. Robert Israel, Table of n, a(n) for n = 1 .. 179 with conjectured 0 values as -1. FORMULA a(n)^2 + n = (n+1)*A350550(n)^2 if a(n) > 0. EXAMPLE a(3) = 0 as the only positive integer solution of p^2 + 3 = 4*q^2 is p=1, q=1, and 1 is not prime. a(4) = 11 as 11^2 + 4 = 125 = (4+1)*5^2 with 11 and 5 prime. MAPLE # Returned values of -1 indicate that either a(n) = 0 or a(n) > 10^1000. f:= proc(n) local m, x, y, S, cf, i, c, a, b, A, M, Sp; m:= n+1; if issqr(m) then S:= [isolve(x^2+n=m*y^2)]; S:= map(t -> subs(t, [x, y]), S); S:= select(t -> andmap(isprime, t), S); if S = [] then return 0 else return min(map(t -> t[1], S)) fi; fi; cf:= NumberTheory:-ContinuedFraction(sqrt(m)); for i from 1 do c:= Convergent(cf, i); if numer(c)^2 - m*denom(c)^2 = 1 then break fi od; a:= numer(c); b:= denom(c); A:= <|>; M:= floor(sqrt(n)*(1+sqrt(a+b*sqrt(m)))/(2*sqrt(m))); S:= select(t -> issqr(m*t^2-m+1), [\$0..M]); S:= select(t -> igcd(t[1], t[2])=1, map(t -> , S)); S:= map(t -> (t, <-t[1], t[2]>), S); if nops(S) = 0 then return 0 fi; for i from 0 do Sp:= select(t -> isprime(t[1]) and isprime(t[2]), S); if nops(Sp)>0 then return min(map(t -> t[1], Sp)) fi; S:= map(t -> A.t, S); if min(map(t -> t[1], S))>10^1000 then break fi; od; -1 end proc: map(f, [\$1..20]); CROSSREFS Cf. A199772, A199773, A350550. Sequence in context: A202350 A096435 A021855 * A256846 A336698 A241837 Adjacent sequences: A350541 A350542 A350543 * A350545 A350546 A350547 KEYWORD nonn,hard,more AUTHOR J. M. Bergot and Robert Israel, Jan 04 2022 STATUS approved

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Last modified March 5 08:03 EST 2024. Contains 370538 sequences. (Running on oeis4.)