OFFSET
1,1
COMMENTS
Conjecture: 2^n-1 is squarefree iff gcd(n,2^n-1)=1. If true, the conjecture would imply that Mersenne numbers (A001348) are squarefree. - Vladeta Jovovic, Apr 12 2002. But the conjecture is not true: counterexamples are n = 364 and n = 1755, i.e., gcd(364,2^364-1) = 1 and (2^364-1) mod 1093^2 = 0 and gcd(1755,2^1755-1) = 1 and (2^1755-1) mod 3511^2 = 0, cf. A001220. - Vladeta Jovovic, Nov 01 2005. The conjecture is true with assumption that n is not a multiple of A002326((q-1)/2), where q is a Wieferich prime A001220. - Thomas Ordowski, Nov 17 2015
If d|n and 2^d-1 is not squarefree, then 2^n-1 cannot be squarefree. For example, if 6 is in the sequence then 6*d is also. - Enrique Pérez Herrero, Feb 28 2009
If p(p-1)|n then p^2|2^n-1, where p is an odd prime. - Thomas Ordowski, Jan 22 2014
The primitive elements of this sequence are A237043. - Charles R Greathouse IV, Feb 05 2014
Dilcher & Ericksen prove that this sequence is exactly the set of numbers divisible by either t(p)p for a Wieferich prime p>2 or t(p) for a non-Wieferich prime p, where t(p) is the order of 2 modulo p (see Proposition 3.1). - Kellen Myers, Jun 09 2015
If d^2 divides 2^n-1 then d divides n, where n is not a multiple of 364, 1755, ...; i.e., A002326((q-1)/2) for Wieferich primes q, A001220. - Thomas Ordowski, Nov 15 2015
(1, 2^n-1, 2^n) is an abc triple iff 2^n-1 is not squarefree. - William Hu, Jul 04 2024
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, A3.
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..296
Karl Dilcher and Larry Ericksen, The Polynomials of Mahler and Roots of Unity, The American Mathematical Monthly, Vol. 122, No. 04 (April 2015), pp. 338-353.
Enrique Pérez Herrero, Mersenne Numbers Treasure Map, Psych Geom blogspot, 02/17/09
Andy Steward, Factorizations of Generalized Repunits [Dead link]
EXAMPLE
a(2)=12 because 2^12 - 1 = 4095 = 5*(3^2)*7*13 is divisible by a square.
MAPLE
N:= 250:
B:= Vector(N):
for n from 1 to N do
if B[n] <> 1 then
F:= ifactors(2^n-1, easy)[2];
if max(seq(t[2], t=F)) > 1 or (hastype(F, symbol)
and not numtheory:-issqrfree(2^n-1)) then
B[[seq(n*k, k=1..floor(N/n))]]:= 1;
fi
fi;
od:
select(t -> B[t]=1, [$1..N]); # Robert Israel, Nov 17 2015
MATHEMATICA
Select[Range[240], !SquareFreeQ[2^#-1]&] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011 *)
PROG
(PARI) default(factor_add_primes, 1);
is(n)=my(f=factor(n>>valuation(n, 2))[, 1], N, o); for(i=1, #f, if(n%(f[i]-1) == 0, return(1))); N=2^n-1; fordiv(n, d, f=factor(2^d-1)[, 1]; for(i=1, #f, if(d==n, return(!issquarefree(N))); o=valuation(N, f[i]); if(o>1, return(1)); N/=f[i]^o)) \\ Charles R Greathouse IV, Feb 02 2014
(PARI) is(n)=!issquarefree(2^n-1) \\ Charles R Greathouse IV, Feb 04 2014
(Magma) [n: n in [1..250] | not IsSquarefree(2^n-1)]; // Vincenzo Librandi, Jul 14 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Apr 12 2002
Definition corrected by Jonathan Sondow, Jun 29 2010
STATUS
approved