A350517 a(n) is the least prime p such that p^2+n = q*(n+1) for some prime q.

3, 2, 3, 19, 5, 41, 3, 19, 11, 67, 5, 131, 13, 41, 17, 101, 17, 37, 11, 29, 109, 47, 5, 101, 53, 1619, 13, 173, 11, 311, 31, 23, 103, 181, 19, 149, 37, 53, 41, 491, 13, 947, 23, 71, 137, 659, 7, 97, 151, 67, 131, 953, 53, 131, 41, 151, 59, 353, 11, 487, 61, 127, 191, 79, 43, 4021, 67, 139, 29

Offset

1,1

Comments

a(n) >= sqrt(n+2), with equality if and only if n+2 is the square of a prime.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n)^2+n = (n+1)*A350518(n).

Example

a(4) = 19 because 19 is prime and 19^2+4 = 365 = 73*(4+1) where 73 is prime, and no smaller prime than 19 works.

Maple

g:= proc(n) local p, M, a, m, q;
M:= sort(map(t -> rhs(op(t)), [msolve(p^2=1, n+1)]));
for a from 0 do
  for m in M do
    p:= a*(n+1)+m;
    if not isprime(p) then next fi;
    q:= (p^2+n)/(n+1);
    if isprime(q) then return p fi
od od:
end proc:
map(g, [$1..100]);

Mathematica

a[n_] := Module[{p = NextPrime[Floor[Sqrt[n + 2]] - 1], q}, While[! IntegerQ [(q = (p^2 + n)/(n + 1))] || ! PrimeQ[q], p = NextPrime[p]]; p]; Array[a, 70] (* Amiram Eldar, Jan 03 2022 *)

Prog

(PARI) isp(r) = (denominator(r)==1) && isprime(r);
a(n) = my(p=2); while (!isp((p^2+n)/(n+1)), p = nextprime(p+1)); p; \\ Michel Marcus, Jan 03 2022
(Python)
from sympy import isprime, nextprime
def A350517(n):
    p = 2
    while True:
        a, b = divmod(p**2+n, n+1)
        if not b and isprime(a):
            return p
        p = nextprime(p) # Chai Wah Wu, Jan 04 2022

Crossrefs

Cf. A350518.

Sequence in context: A298594 A092950 A059239 * A123170 A253381 A091806

Adjacent sequences: A350514 A350515 A350516 * A350518 A350519 A350520

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Jan 02 2022

Status

approved