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%I #8 Apr 20 2023 08:24:01
%S 1,1,2,3,5,4,3,7,2,3,5,4,3,7,2,3,5,4,3,7,2,3,5,4,3,7,2,3,5,4,3,7,2,3,
%T 5,4,3,7,2,3,5,4,3,7,2,3,5,4,3,7,2,3,5,4,3,7,2,3,5,4,3,7,2,3,5,4,3,7,
%U 2,3,5,4,3,7,2,3,5,4,3,7,2,3,5,4,3,7,2,3,5,4,3,7,2,3,5,4,3,7,2,3,5,4,3,7,2
%N a(n) = a(n-1) + a(n-2); if a(n) is not prime divide a(n) by its largest prime factor.
%C Sequence repeats cyclically. Conjecture: sequence ends with cycle for any positive a(1) and a(2). This has been checked for a(1)=1 and a(2)=1,...,2000. Some other cases shown below: a(1), a(2), length of s, sequence s 1,1,10,{1,1,2,3,5,4,3,7,2,3}, 1,2,9,{1,2,3,5,4,3,7,2,3}, 1,3,9,{1,3,2,5,7,4,11,3,2}, 1,4,20,{1,4,5,3,4,7,11,6,17,23,8,31,3,2,5,7,4,11,3,2}, 1,5,12,{1,5,2,7,3,2,5,7,4,11,3,2}, 1,6,23,{1,6,7,13,4,17,3,4,7,11,6,17,23,8,31,3,2,5,7,4,11,3,2}, 1,7,9,{1,7,4,11,3,2,5,7,4}, 1,8,30,{1,8,3,11,2,13,3,8,11,19,6,5,11,8,19,9,4,13,17,6,23,29,4,3,7,2,3,5,4,3}, 1,9,34,{1,9,2,11,13,8,3,11,2,13,3,8,11,19,6,5,11,8,19,9,4,13,17,6,23,29,4,3,7,2,3,5,4,3}, 1,10,10,{1,10,11,3,2,5,7,4,11,3}. Even exotic seeds give no long sequences, e.g. 967,3000,24,{967,3000,3967,6967,154,7121,75,28,103,131,18,149,167,4,9,13,2,3,5,4,3,7,2,3}. For a(1)=1, maximal length = 55 is found for a(2)=1207: 1,1207,55,{1,1207,8,243,251,26,277,3,40,43,83,18,101,7,36,43,79,2,27,29,8,37,9,2,11,13,8,3,11,2,13,3,8,11,19,6,5,11,8,19,9,4,13,17,6,23,29,4,3,7,2,3,5,4,3}.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,1).
%F G.f.: -(6*x^8+2*x^7+4*x^6+5*x^5+3*x^4+2*x^3+x^2+x)/(x^6-1). - _Alois P. Heinz_, Apr 20 2023
%Y Cf. A214674.
%K nonn,easy
%O 1,3
%A _Zak Seidov_, Mar 09 2006
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