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

1,1

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

Links

Table of n, a(n) for n=1..62.

Example

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

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 *)

Prog

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

Crossrefs

Cf. A005117, A049094, A049096.

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

Sequence in context: A186455 A318081 A276655 * A318349 A228142 A276756

Adjacent sequences: A187962 A187963 A187964 * A187966 A187967 A187968

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Mar 18 2011

Extensions

More terms from Joerg Arndt, Nov 23 2015

Status

approved