A350518 - OEIS

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A350518

a(n) is the least prime q such that q*(n+1)-n is the square of a prime.

5, 2, 3, 73, 5, 241, 2, 41, 13, 409, 3, 1321, 13, 113, 19, 601, 17, 73, 7, 41, 541, 97, 2, 409, 109, 97081, 7, 1033, 5, 3121, 31, 17, 313, 937, 11, 601, 37, 73, 43, 5881, 5, 20857, 13, 113, 409, 9241, 2, 193, 457, 89, 331, 17137, 53, 313, 31, 401, 61, 2113, 3, 3889, 61, 257, 571, 97, 29, 241321

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OFFSET

1,1

LINKS

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

FORMULA

a(n)*(n+1)-n = A350517(n)^2.

EXAMPLE

a(4) = 73 because 73 is prime, 73*(4+1)-4 = 361 = 19^2 where 19 is prime, and no smaller prime than 73 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 q fi

od od:

end proc:

map(g, [$1..100]);

MATHEMATICA

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

PROG

(PARI) a(n) = my(q=2, p); while(! (issquare(q*(n+1)-n, &p) && isprime(p)), q = nextprime(q+1)); q; \\ Michel Marcus, Jan 03 2022

(Python)

from sympy import integer_nthroot, isprime, nextprime

def A350518(n):

q = 2

while True:

a, b = integer_nthroot(q*(n+1)-n, 2)

if b and isprime(a):

return q

q = nextprime(q) # Chai Wah Wu, Jan 04 2022

CROSSREFS