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A113910 Integers of the form (Lucas(i+1) - 2*A006206(i+2))/(A006206(i+2) - A006206(i)), i > 2; Lucas = A000204.

%I A113910 #7 Sep 03 2017 03:24:47
%S A113910 3,7,5,9,11,17,29,41,59,71,101,107,137,149,179,191,197,227,239,269,
%T A113910 281,311,347,419,431,461,521,569,599,617,641,659,809,821,827,857,881,
%U A113910 1019,1031,1049,1061,1091,1151,1229,1277,1289,1301,1319,1427,1451,1481,1487
%N A113910 Integers of the form (Lucas(i+1) - 2*A006206(i+2))/(A006206(i+2) - A006206(i)), i > 2; Lucas = A000204.
%C A113910 Let p and p+2 be twin primes. Then Lucas(p) = 1 + p*A006206(p) and Lucas(p+2) = 1 + (p+2)*A006206(p+2). It follows from Lucas(n) + Lucas(n+1) = Lucas(n+2) that p = (Lucas(p+1) - 2*A006206(p+2))/(A006206(p+2) - A006206(p))
%C A113910 For i = 3, 4, 5, 6, 7, 8, 9, 10, 11: ((Lucas(i+1) - 2*A006206(i+2))/(A006206(i+2) - A006206(i))) = (3, 7, 5, 19/3, 31/4, 9, 87/10, 149/14, 11, 135/11, 663/50, 1094/77, 1787/120, 2939/181, 17, 7849/434, 12799/672, 20894/1041, 34031/1622, 55469/2514, 45131/1962, 146921/6115, 238915/9554, 194252/7465, 631347/23386, 1025917/36617, 29, 2706059/90178, 4393211/141710, 3565643/111405, 11573003/350702). - _Creighton Dement_, Jan 31 2006
%F A113910 It is conjectured that a(n+4) = A001359(n+2) for all n.
%p A113910 # First 63 Terms with(combinat): with(numtheory): A006206 := proc(n) local sum; sum := 0; for d in divisors(n) do sum := sum + mobius(n/d)*(fibonacci(d+1)+fibonacci(d-1)) od; RETURN(sum/n); end; A000204 := n->fibonacci(n+1)+fibonacci(n-1); T := n -> (A000204(n+1) - 2*A006206(n+2))/(A006206(n+2)-A006206(n)); A113910 := []: for i from 3 by 1 to 2000 do if is(T(i) = floor(T(i))) then A113910 := [op(A113910), T(i)]; fi: od: A113910; # _Creighton Dement_, Jan 15 2009
%Y A113910 Cf. A000204, A006206, A001359.
%K A113910 nonn
%O A113910 1,1
%A A113910 _Creighton Dement_, Jan 29 2006
%E A113910 Extended and Maple definition by _Creighton Dement_, Jan 15 2009