A387642 a(n) is the least number x such that x^2 + 1 and 2^x + 1 are both divisible by A387595(n).

0, 2, 18, 4, 70, 450, 50, 182, 1170, 18, 216, 4050, 1602, 98, 374, 742, 4514, 286, 1118, 12870, 882, 6288, 16562, 1710, 900, 2040, 182, 14070, 3818, 5005, 2482, 16614, 14062, 13398, 748, 30690, 5754, 30846, 1254, 32100, 16626, 76230, 612, 20272, 2394, 83950, 450, 23622, 54470, 5790, 117990, 1602

Offset

1,2

Comments

a(n) is the least number x such that x^2 + 1 and 2^x + 1 are both divisible by the n-th number k for which there exists x such that x^2 + 1 and 2^x + 1 are both divisible by k.

Links

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

Formula

a(n)^2 == 2^a(n) == -1 (mod A387595(n)).

Example

a(3) = 18 because A387595(3) = 13, 18^2 == 2^18 == -1 (mod 13), and no smaller number works.

Maple

f:= proc(n) local a, b, x, y, t, tt;
    uses NumberTheory;
    if n::even then return -1 fi;
    a:=[msolve(x^2 + 1 = 0, n)];
    if a = [] then return -1 fi;
    a:= map(t -> rhs(op(t)), a);
    b:=msolve(2^x + 1 = 0, t, n);
    if b = {} or b = NULL then return -1 fi;
    b:= rhs(op(b));
    tt:= indets(b)[1];
    subs(infinity=-1, min(map(proc(s) local r; r:= ChineseRemainder([s, subs(tt=0, b)], [n, coeff(b, tt)]); if r = FAIL then +infinity else r fi end proc, a)));
end proc:
f(1):= 0:
subs(-1=NULL, map(f, [$1..1000]));

Crossrefs

Cf. A066787, A387595.

Sequence in context: A383240 A253603 A094048 * A179073 A087665 A093481

Adjacent sequences: A387639 A387640 A387641 * A387643 A387644 A387645

Keyword

nonn

Author

Robert Israel, Sep 04 2025

Status

approved