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%I #34 Oct 29 2023 21:41:35
%S 341,1105,1387,1729,2047,2465,2701,2821,3277,4033,4369,4681,5461,6601,
%T 7957,8321,8911,10261,10585,11305,13741,13747,13981,14491,15709,15841,
%U 16705,18721,19951,23377,29341,30121,30889,31417,31609,31621,34945
%N Composite numbers k which divide A001045(k-1).
%C Also composite numbers k such that (2^k - 2)/3 + 1 == 2^k - 1 == 1 (mod k).
%C An equivalent definition of this sequence: pseudoprimes to base 2 that are not divisible by 3. - _Arkadiusz Wesolowski_, Nov 15 2011
%C Conjecture: these are composites k such that 2^M(k-1) == -1 (mod M(k)^2 + M(k) + 1), where M(k) = 2^k - 1. - _Amiram Eldar_ and _Thomas Ordowski_, Dec 19 2019
%C These are composites k such that 2^(m-1) == 1 (mod (m+1)^6 - 1), where m = 2^k - 1. - _Thomas Ordowski_, Sep 17 2023
%H Charles R Greathouse IV, <a href="/A066488/b066488.txt">Table of n, a(n) for n = 1..10000</a>
%t a[0] = 0; a[1] = 1; a[n_] := a[n] = a[n - 1] + 2a[n - 2]; Select[ Range[50000], IntegerQ[a[ # - 1]/ # ] && !PrimeQ[ # ] && # != 1 & ]
%t fQ[n_] := ! PrimeQ@ n && Mod[((2^n - 2)/3 + 1), n] == Mod[2^n - 1, n] == 1; Select[ Range@ 35000, fQ]
%o (PARI) is(n)=n%3 && Mod(2,n)^(n-1)==1 && !isprime(n) && n>1 \\ _Charles R Greathouse IV_, Sep 18 2013
%o (Magma) [k:k in [4..40000]|not IsPrime(k) and ((2^(k-1) + (-1)^k) div 3) mod k eq 0]; // _Marius A. Burtea_, Dec 20 2019
%Y Cf. A001567, A002808, A001045, A066047.
%K easy,nonn
%O 1,1
%A _Robert G. Wilson v_, Jan 03 2002
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