A372457 a(n) is the least k such that k^2 + k + 1 is divisible by the n-th power of a prime.

1, 18, 18, 1047, 1353, 34967, 82681, 2387947, 14906455, 135967276, 700917774, 4655571260, 18496858461, 272170172759, 950393245608, 10445516265494, 43678446835095, 654213095126525, 654213095126525, 22143577275619760, 101935843573231761, 1777573435823083782, 6042068661342892315

Offset

1,2

Comments

For n > 1 the prime is in A002476. Conjecture: it is always 7.

Links

Robert Israel, Table of n, a(n) for n = 1..1182

Example

a(4) = 1047 because 1047^2 + 1047 + 1 = 1097257 is divisible by 7^4.

Maple

g:= proc(n) local p, t, tm, r, s, S;
     tm:= infinity; r:= infinity;
     for p from 7 by 6 do
       if p^n > r then return tm fi;
       if not isprime(p) then next fi;
       S:= [msolve(t^2+t+1, p^n)];
       if S = [] then next fi;
       s:= min(map(rhs@op, S));
       if s < tm then tm:= s; r:= s^2 + s + 1 fi;
     od;
end proc:
g(1):= 1:
map(g, [$1..30]);

Prog

(Python)
from sympy import sqrt_mod_iter, nextprime
def A372457(n):
    if n == 1: return 1
    p, m, r = 7, None, None
    while (m is None or p**n <= m):
        if (k:=min((r>>1 for r in sqrt_mod_iter(-3, p**n) if r&1), default=None)) is not None:
            m = (r:=k if r is None else min(r, k))*(r+1)+1
        while (p:=nextprime(p))%6!=1: pass
    return r # Chai Wah Wu, May 02 2024

Crossrefs

Cf. A002476.

Sequence in context: A352365 A050686 A165839 * A177014 A355045 A357800

Adjacent sequences: A372454 A372455 A372456 * A372458 A372459 A372460

Keyword

nonn

Author

Robert Israel, May 01 2024

Status

approved