A359437 a(n) is the first prime p such that there are exactly n numbers i with 1 <= i < p such that one of i*p-(p-i) and i*p+(p-i) is a prime and the other is the square of a prime.

2, 17, 11, 7, 239, 167, 1933, 9241, 19319, 120121, 649991, 4564559, 513239, 11324041, 31831799, 54708721, 59219161, 215975759, 241431959, 265012441, 549789239, 138389159, 3336693359, 1990674841

Offset

0,1

Comments

Suggested in an email from J. M. Bergot.

It appears that in most cases the squares are either all i*p-(p-i) or all i*p+(p-i). However, this is not the case for a(3) or a(18).

a(21) = 138389159.

Links

Table of n, a(n) for n=0..23.

Example

a(1) = 17 because 2*17 - (17 - 2) =  19, 2*17 + (17 - 2) = 7^2.
a(2) = 11 because 3*11 - (11 - 3) = 5^2, 3*11 + (11 - 3) =  41;
                  5*11 - (11 - 5) = 7^2, 5*11 + (11 - 5) =  61.
a(3) =  7 because 2*7  -  (7 - 2) = 3^2, 2*7  +  (7 - 2) =  19;
                  3*7  -  (7 - 3) =  17, 3*7  +  (7 - 3) = 5^2;
                  4*7  -  (7 - 4) = 5^2, 4*7  +  (7 - 4) =  31.

Maple

f:= proc(p) local x, S1, S2, R1, R2;
S1:= {msolve(x^2 = -p, p+1)};
R1:= select(i -> i < p and isprime(i*p+p-i), map(t -> (t^2+p)/(p+1), select(isprime, map(rhs@op, S1))));
S2:= {msolve(x^2 = p, p-1)};
R2:= select(i -> i < p and isprime(i*p-p+i), map(t -> (t^2-p)/(p-1), select(isprime, map(rhs@op, S2))));
nops(R1 union R2)
end proc:
f(2):= 0:
V:= Array(0..12): count:= 0:
p:= 1:
while count < 13 do
p:= nextprime(p); v:= f(p);
if V[v] = 0 then V[v]:= p; count:= count+1 fi
od:
convert(V, list);

Prog

(Python)
from sympy import sqrt_mod_iter, nextprime, isprime
def A359437(n):
    p = 1
    while (p:=nextprime(p)):
        if len(set(filter(lambda x:isprime(p*(x+1)-x), ((d**2+p)//(p+1) for d in sqrt_mod_iter(-p, p+1) if isprime(d)))) | set(filter(lambda x: isprime(p*(x-1)+x), ((d**2-p)//(p-1) for d in sqrt_mod_iter(p, p-1) if isprime(d)))))==n:
            return p # Chai Wah Wu, May 06 2024

Crossrefs

Sequence in context: A362397 A257466 A226291 * A077311 A196732 A346391

Adjacent sequences: A359434 A359435 A359436 * A359438 A359439 A359440

Keyword

nonn,more

Author

Robert Israel, Dec 31 2022

Extensions

a(21) from Robert Israel, Dec 31 2022

a(20), a(22)-a(23) from Chai Wah Wu, May 06 2024

Status

approved