A260119 a(n) is the least positive integer k such that 2^n-1 and k^n-1 are relatively prime.

1, 3, 3, 15, 3, 21, 3, 45, 3, 99, 5, 1365, 3, 3, 3, 765, 3, 399, 3, 1815, 3, 69, 5, 1365, 3, 3, 3, 435, 3, 35805, 3, 765, 5, 3, 7, 2878785, 3, 3, 3, 20295, 3, 903, 5, 1035, 3, 141, 3, 116025, 3, 99, 3, 795, 3, 399, 5, 435, 3, 177, 3, 85180095, 3, 3, 3, 765, 3, 32361, 3, 45, 5, 11715, 3

Offset

1,2

Comments

Note that (2^n-1)^n-1 is always relatively prime to 2^n-1.

a(72) > 10^8.

According to the conjecture given in A086892, a(n) = 3 infinitely often.

From David A. Corneth, Aug 17 2015: (Start)

Conjecture 1: a(n) is never divisible by 2.

Conjecture 2: a(n) is of the form q * m where q is the product of all odd primes that are one more than a divisor of n.

(End)

From Robert Israel, Sep 02 2015: (Start)

Both conjectures are true.

Since (2k)^n - 1 = (k^n)*(2^n - 1) + (k^n - 1), GCD((2k)^n - 1, 2^n-1) = GCD(k^n-1, 2^n-1). Thus a(n) is always odd.

If p is an odd prime such that p-1 divides n, then k^n - 1 is divisible by p for all k coprime to p (and in particular k=2). Thus a(n) must be divisible by p, and thus by the product of all such p.

(End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000 [Terms 72 through 5000 were computed by David A. Corneth, Aug 17 2015]

Example

Since 2^5-1 = 31 and 3^5-1 = 242 are relatively prime, a(5) = 3.
The divisors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72;
Adding one to each divisor gives 3, 4, 5, 7, 9, 10, 13, 19, 25, 37, 73;
The odd primes in this list are 3, 5, 7, 13, 19, 37, 73;
The product of these primes is 3 * 5 * 7 * 13 * 19 * 37 * 73 = 70050435;
Thus a(72) is of the form 70050435 * m.

Maple

f:= proc(n) local P, Q, M, k, kn;
P:= select(isprime, map(t -> t+1, numtheory:-divisors(n)) minus {2});
M:= convert(P, `*`);
Q:= 2^n - 1;
for k from M by 2*M do
  kn:= k &^ n - 1 mod Q;
  if igcd(kn, Q) = 1 then
    return k
  fi
od
end proc:
map(f, [$1..100]); # Robert Israel, Sep 02 2015

Mathematica

Table[k = 1; While[! CoprimeQ[2^n - 1, k^n - 1], k++]; k, {n, 59}] (* Michael De Vlieger, Sep 01 2015 *)

Prog

(Sage)
def min_k(n):
    g, k=2, 0
    while g!=1:
        k=k+1
        g=gcd(2^n-1, k^n-1)
    return k
print([min_k(n) for n in [1..71]])
(PARI) a(n) = {my(k=1, pt = 2^n-1); while (gcd(pt, k^n-1) != 1, k++); k; } \\ Michel Marcus, Jul 17 2015 && Jan 27 2016
(PARI) conjecture_a(n) = {my(d=divisors(n)); v=select(x->isprime(x+1), select(y->y%2==0, d)); v+= vector(#v, i, 1); p=prod(i=1, #v, v[i]); if(p==1&&n!=1, p=3);
forstep(i=1, p, 2, if(gcd((p * i)^n-1, 2^n-1)==1, return(p*i))); for(i=1, n, if(gcd(p^n-1, 2^n-1) == 1, return(p), p=nextprime(p+1))); forstep(i=1, 100000, 2, if(gcd((i)^n-1, 2^n-1)==1, return(i)))} \\ David A. Corneth, Sep 01 2015

Crossrefs

Cf. A086892.

Sequence in context: A282388 A131943 A226139 * A368117 A282009 A282485

Adjacent sequences: A260116 A260117 A260118 * A260120 A260121 A260122

Keyword

nonn

Author

Tom Edgar, Jul 17 2015

Status

approved