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 A066488 Composite numbers n which divide A001045(n-1). 2

%I

%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 n which divide A001045(n-1).

%C Also composite numbers n such that ((2^n - 2)/3 + 1) == 2^n - 1 == 1 (mod n).

%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 composite numbers n such that 2^M(n-1) == -1 (mod M(n)^2 + M(n) + 1), where M(n) = 2^n - 1. - _Amiram Eldar_ and _Thomas Ordowski_, Dec 19 2019

%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. A001045, A066047.

%K easy,nonn

%O 1,1

%A _Robert G. Wilson v_, Jan 03 2002

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Last modified June 6 11:14 EDT 2020. Contains 334827 sequences. (Running on oeis4.)