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A216231 Fibonacci with priority for primes: a(0)=0, a(1)=1, for n >= 2, a(n) = a(n-1) + a(k), where 0 < k <= n-2 is maximal index such that a(n-1) + a(k) is prime. If there is no such k, then a(n) = a(n-1) + a(n-2).

%I A216231 #36 Feb 07 2025 09:26:24
%S A216231 0,1,1,2,3,5,7,12,19,31,43,74,79,153,227,239,313,552,631,643,1274,
%T A216231 1427,1979,3253,5232,7211,7213,14424,14737,15289,20521,20533,41054,
%U A216231 41281,82335,83609,83621,88853,90127,104551,194678,201889,207121,212353,226777,226789
%N A216231 Fibonacci with priority for primes: a(0)=0, a(1)=1, for n >= 2, a(n) = a(n-1) + a(k), where 0 < k <= n-2 is maximal index such that a(n-1) + a(k) is prime. If there is no such k, then a(n) = a(n-1) + a(n-2).
%C A216231 Conjecture: There exist arbitrarily long chains of consecutive prime terms.
%C A216231 Agrees with the lower matching number of the n-Fibonacci cube graph from n = 1 to at least n = 9. - _Eric W. Weisstein_, Feb 07 2025
%H A216231 Peter J. C. Moses, <a href="/A216231/b216231.txt">Table of n, a(n) for n = 0..9999</a>
%p A216231 a:= proc(n) option remember; local k;
%p A216231       if n<2 then n
%p A216231     else for k from n-2 to 1 by -1
%p A216231            while not isprime(a(n-1) +a(k)) do od;
%p A216231          a(n-1) +a(`if`(k=0, n-2, k))
%p A216231       fi
%p A216231     end:
%p A216231 seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 14 2013
%t A216231 a216231[0]:=0;
%t A216231 a216231[1]:=1;
%t A216231 a216231[n_]:=a216231[n]=
%t A216231 Module[{k},(k=NestWhile[#-1&,n-1,(#>1)&&!PrimeQ[a216231[n-1]+a216231[#]]&];
%t A216231 If[k==1,k=n-2]);a216231[n-1]+a216231[k]];
%t A216231 Table[a216231[n],{n,0,100}] (* _Peter J. C. Moses_, Mar 14 2013 *)
%Y A216231 Cf. A000045.
%K A216231 nonn
%O A216231 0,4
%A A216231 _Vladimir Shevelev_, Mar 14 2013