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A176033 Numbers k such that 2^(2k-1) == 2 (mod 2k) and such that 2^(k-1) != 1 (mod k).

%I A176033 #37 Jul 07 2025 14:53:47
%S A176033 15,85,91,435,451,703,1247,1271,1581,1695,1891,2071,3133,3367,3683,
%T A176033 4795,4859,5551,6643,8695,9061,9131,9211,9605,9919,12403,13019,14351,
%U A176033 14701,15051,15211,16021,16471,19669,20191,20485,24727,25351,26335,26599,27511,28645
%N A176033 Numbers k such that 2^(2k-1) == 2 (mod 2k) and such that 2^(k-1) != 1 (mod k).
%C A176033 The associated value m = (2^(k-1) mod k) satisfy 1 < gcd(m-1, k) < k.
%C A176033 The selection criterion 2^(2k-1) == 2 (mod 2k) admits 3, 5, 7, 11, 13, 15, 17, etc.
%C A176033 Expect that the sequences will be infinite only if the criterion has the form 2^(2k-1) == 2^m (mod 2k) where m - an integer (1, 2, ...), otherwise the sequence will be limited. For example, for criterion 2^(2k-1) == 14 (mod 2k), the sequence begins 9, 27, 7281, 19143.
%H A176033 Robert Israel, <a href="/A176033/b176033.txt">Table of n, a(n) for n = 1..1000</a>
%p A176033 select(n -> 2 &^ (2*n-1) - 2 mod (2*n) = 0 and 2 &^ (n-1) -1 mod n <> 0, [seq(n,n=3..10^5,2)]); # _Robert Israel_, Nov 06 2017
%t A176033 Select[Range[30000],PowerMod[2,2#-1,2#]==2&&PowerMod[2,#-1,#]!=1&] (* _Harvey P. Dale_, Jul 06 2025 *)
%o A176033 (PARI) alist(m) = {for (n=1, m, v = 2^(2*n-1); if ((v % (2*n) == 2), k = 2^(n-1) % n; if (k > 1, print1(n, ", "););););} \\ _Michel Marcus_, Jan 24 2013
%Y A176033 Set difference of A020136 and A001567. - _Max Alekseyev_, Nov 06 2017
%K A176033 nonn
%O A176033 1,1
%A A176033 _Alzhekeyev Ascar M_, Dec 06 2010
%E A176033 More terms from _Michel Marcus_, Jan 24 2013