A350517 - OEIS

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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

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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