A049093 Numbers n such that 2^n - 1 is squarefree.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

Numbers n such that gcd(n, 2^n - 1) = 1 and n is not a multiple of A002326((q - 1)/2), where q is a Wieferich prime A001220. - Thomas Ordowski, Nov 21 2015

If n is in the sequence, then so are all divisors of n. - Robert Israel, Nov 23 2015

Examples

			a(7) = 8 because 2^8 - 1 = 255 = 3 * 5 * 17 is squarefree.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..910

Crossrefs

Complement of A049094.

Programs

Magma

[n: n in [1..100] | IsSquarefree(2^n-1)]; // Vincenzo Librandi, Nov 22 2015

Maple

N:= 400: # to get all terms <= N
# This relies on the fact that the first N+1 members of A000225 have all been factored
# without any further Wieferich primes being found.
V:= Vector(N,1):
V[364 * [$1..N/364]]:= 0:
V[1755 * [$1..N/1755]]:= 0:
for n from 2 to N do
if V[n] = 0 then next fi;
if igcd(n, 2 &^n - 1 mod n) > 1 then
  V[n * [$1..N/n]]:= 0
fi;
od:
select(t -> V[t] = 1, [$1..N]); # Robert Israel, Nov 23 2015

Mathematica

Select[Range@ 92, SquareFreeQ[2^# - 1] &] (* Michael De Vlieger, Nov 21 2015 *)

PARI

isok(n) = issquarefree(2^n - 1); \\ Michel Marcus, Dec 19 2013

Extensions

Terms a(73)-a(910) in b-file from Max Alekseyev, Nov 15 2014, Sep 28 2015