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A139791 Numbers n for which 2n is a multiple of A002326(n), the multiplicative order of 2 mod 2n+1.

%I A139791 #22 Oct 14 2022 21:50:10
%S A139791 1,2,3,5,6,8,9,11,14,15,18,20,21,23,26,29,30,33,35,36,39,41,44,48,50,
%T A139791 51,53,54,56,63,65,68,69,74,75,78,81,83,86,89,90,95,96,98,99,105,111,
%U A139791 113,114,116,119,120,125,128,131,134,135,138,140,141,146,153,155,156,158,165,168,170
%N A139791 Numbers n for which 2n is a multiple of A002326(n), the multiplicative order of 2 mod 2n+1.
%C A139791 The sequence properly contains A005097. 170 is the first number which is not in A005097. One can prove that A002326(2^(2t-1)) = 4t. Thus if n=2^(2t-1), where, for any m>0, t=2^(m-1) then 2n is a multiple of A002326(n) while 2n+1 is a Fermat number which, as well known, is not always a prime.
%C A139791 The sequence is the union of A005097 and (A001567 - 1)/2. [Conjectured by _Vladimir Shevelev_, proved by _Ray Chandler_, May 26 2008]
%D A139791 Christopher Adler and Jean-Paul Allouche (2022), Finite self-similar sequences, permutation cycles, and music composition, Journal of Mathematics and the Arts, 16:3, 244-261, DOI: 10.1080/17513472.2022.2116745.
%H A139791 Amiram Eldar, <a href="/A139791/b139791.txt">Table of n, a(n) for n = 1..10000</a>
%t A139791 Select[Range[160], Divisible[2#, MultiplicativeOrder[2, 2#+1]] &] (* _Amiram Eldar_, Jun 28 2019 *)
%o A139791 (PARI) isok(n) = !(2*n % znorder(Mod(2, 2*n+1))); \\ _Michel Marcus_, Nov 02 2017
%Y A139791 Cf. A002326, A005097, A001262, A001567, A137576.
%K A139791 nonn
%O A139791 1,2
%A A139791 _Vladimir Shevelev_, May 21 2008, May 24 2008
%E A139791 Data extended up to a(68) = 170 to clarify distinction from A005097 and essentially identical sequences A130290 and A102781, by _M. F. Hasler_, Dec 13 2019