A187965 Numbers k such that 2^k - 1 and 2^k + 1 are not squarefree.

21, 30, 63, 78, 90, 105, 110, 147, 150, 189, 204, 210, 231, 234, 270, 273, 310, 315, 330, 340, 357, 390, 399, 441, 450, 465, 483, 510, 525, 546, 550, 567, 570, 609, 612, 630, 651, 657, 666, 690, 693, 702, 735, 750, 759, 770, 777, 810, 819, 858, 861, 870, 903, 930, 945, 987, 990, 1014, 1020, 1029, 1050, 1071

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Mar 18 2011

Keywords

• nonn

Comments

If k is in the sequence, then so is m*k for any odd m. - Thomas Ordowski, Nov 23 2015

Note that 110, 310, 340, 550, 770 are not divisible by 3.

Let b(p) be the multiplicative order of 2 modulo p^2. Then k is in this sequence if and only if there exists odd primes p, q such that b(p) | k and k == b(q)/2 (mod b(q)) with even b(q). For example, we have b(7) = 21, b(3) = 6 so b(7) | 21, 21 == b(3)/2 (mod b(3)), hence 21 is a term; likewise, b(3) = 6, b(5) = 20, so b(3) | 30, 30 == b(5)/2 (mod b(5)), hence 30 is a term. - Jianing Song, Jan 20 2021

Examples

			2^21 - 1 = 7^2 * 127 * 337, 2^21 + 1 = 3^2 * 43 * 5419.
		

Crossrefs

Cf. A005117, A049094, A049096.

Cf. A243905 (multiplicative orders of 2 modulo p^2), A242777 (k+1 is prime).

Programs

Magma

[n: n in [1..250] | not IsSquarefree(2^n-1) and not IsSquarefree(2^n+1)]; // Vincenzo Librandi, Nov 23 2015

Mathematica

Select[ Range@ 500, !(SquareFreeQ[2^# - 1] || SquareFreeQ[2^# + 1]) &]
Select[Range[1100],NoneTrue[2^#+{1,-1},SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 22 2019 *)

PARI

is(n) = !issquarefree(2^n-1) && !issquarefree(2^n+1);
for(n=1, 1e3, if(is(n), print1(n, ", "))) \\ Altug Alkan, Nov 22 2015

Extensions

More terms from Joerg Arndt, Nov 23 2015