A049094 Numbers m such that 2^m - 1 is divisible by a square > 1.

6, 12, 18, 20, 21, 24, 30, 36, 40, 42, 48, 54, 60, 63, 66, 72, 78, 80, 84, 90, 96, 100, 102, 105, 108, 110, 114, 120, 126, 132, 136, 138, 140, 144, 147, 150, 155, 156, 160, 162, 168, 174, 180, 186, 189, 192, 198, 200, 204, 210, 216, 220, 222, 228, 231, 234, 240

Offset: 1

Author

Labos Elemer

Keywords

• nonn

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

Examples

			a(2)=12 because 2^12 - 1 = 4095 = 5*(3^2)*7*13 is divisible by a square.
		

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]

Crossrefs

Cf. A001220, A001348, A002326, A014491, A049093, A049096, A049095, A237043, A282242, A282243.

Programs

Magma

[n: n in [1..250] | not IsSquarefree(2^n-1)]; // Vincenzo Librandi, Jul 14 2015

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 *)

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

Extensions

More terms from Vladeta Jovovic, Apr 12 2002

Definition corrected by Jonathan Sondow, Jun 29 2010