cp's OEIS Frontend

A237043 Numbers n such that 2^n - 1 is not squarefree, but 2^d - 1 is squarefree for every proper divisor d of n.

%I A237043 #48 Aug 15 2025 12:48:40
%S A237043 6,20,21,110,136,155,253,364,602,657,812,889,979,1081
%N A237043 Numbers n such that 2^n - 1 is not squarefree, but 2^d - 1 is squarefree for every proper divisor d of n.
%C A237043 Primitive elements of A049094: the elements of A049094 are precisely the positive multiples of members of this sequence.
%C A237043 If p^2 divides 2^n - 1 for some odd prime p, then by definition the multiplicative order of 2 mod p^2 divides n. The multiplicative order of 2 mod p^2 is p times the multiplicative order of 2 mod p unless p is a Wieferich prime, in which case the two orders are identical. Hence either p is a Wieferich prime or p*log_2(p+1) <= n. This should allow finding larger members of this sequence. - _Charles R Greathouse IV_, Feb 04 2014
%C A237043 If n is in the sequence and m>1 then m*n is not in the sequence. Because n is a proper divisor of m*n and 2^n-1 is not squarefree. - _Farideh Firoozbakht_, Feb 11 2014
%C A237043 a(15) >= 1207. - _Max Alekseyev_, Sep 28 2015
%C A237043 From _Daniel Suteu_, Jul 03 2019: (Start)
%C A237043 The following numbers are also in the sequence: {1755, 2265, 2485, 2756, 3081, 3164, 4112, 6757, 8251, 13861, 18533}.
%C A237043 Probably, the following numbers are also terms: {3422, 5253, 6806, 8164, 9316, 11342, 12550, 15025, 15026, 17030, 17404, 17468, 18145, 19670, 19701, 22052}. (End)
%H A237043 Peice Hua, <a href="https://arxiv.org/abs/2508.05981">Finite 2-groups having a cyclic or dihedral maximal subgroup and arc-transitive maps</a>, arXiv:2508.05981 [math.GR], 2025. See p. 3.
%t A237043 Select[Range@ 160, And[AllTrue[2^#2 - 1, SquareFreeQ], ! SquareFreeQ[2^First@ #1 - 1]] & @@ TakeDrop[Divisors@ #, -1] &] (* _Michael De Vlieger_, Jul 07 2019 *)
%o A237043 (PARI) default(factor_add_primes, 1);
%o A237043 isA049094(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))
%o A237043 is(n)=fordiv(n,d,if(isA049094(d),return(d==n))); 0
%o A237043 (PARI) \\ Simpler but slow
%o A237043 is(n)=fordiv(n,d,if(!issquarefree(2^d-1),return(d==n))); 0
%Y A237043 Cf. A049094, A005420, A065069, A282631, A282632.
%K A237043 nonn,hard,more
%O A237043 1,1
%A A237043 _Charles R Greathouse IV_, Feb 02 2014
%E A237043 a(14) from _Charles R Greathouse IV_, Sep 21 2015, following Womack's factorization of 2^991-1.