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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
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:= <<a, b>|<m*b, 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 -> <sqrt(m*t^2-m+1), 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
Sequence in context: A202350 A096435 A021855 * A256846 A336698 A241837
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.)