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%I #20 Jan 21 2019 19:57:41
%S 11,10,11,4,1,9,11,2,11,12,11,2,2,12,11,12,3,2,11,7,11,2,11,12,11,12,
%T 11,2,4,4,11,2,11,3,11,12,8,2,11,12,43,2,11,3,11,5,11,2,11,44,11,2,11,
%U 12,11,12,11,2,11,3,11,2,11,4,11,12,11,2,11,3,11,2,11,12,11,12,11,2,11,9,11,2,11,12,11,3,11,2,11,12,11,2,11,12,11,3,11,2,11,12,11,2,11,5,11,4,11,2,11
%N a(n) is one less than the length of the sequence b(n) defined by: b(1) = n; for k > 1, b(k+1) is the smallest prime factor of 1+b(k)^2 not already in the b-sequence.
%C The number of terms of each sequence b(n) is finite.
%C Records are a(1)=11, a(10)=12, a(41)=43, a(50)=44, and apparently no others; i.e., the next is not smaller than n=360000 if it exists.
%e a(1) = 11 counts the iterations in the b-chain 1 -> 2 -> 5 -> 13 -> 17 -> 29 -> 421 -> 401 -> 37 -> 137 -> 1877 -> 41 (end of the cycle because 41 -> 29). In detail the chain is computed as follows:
%e 1^2 + 1 = 2;
%e 2^2 + 1 = 5;
%e 5^2 + 1 = 2*13 -> 13 because 2 is already in the sequence;
%e 13^2 + 1 = 2*5*17 -> 17 because 2 and 5 are already in the sequence;
%e 17^2 + 1 = 2*5*29 -> 29;
%e 29^2 + 1 = 2*421 -> 421;
%e 421^2 + 1 = 2*13*17*401 -> 401;
%e 401^2 + 1 = 2*37*41*53 -> 37;
%e 37^2 + 1 = 2*5*137 -> 137;
%e 137^2 + 1 = 2*5*1877 -> 1877;
%e 1877^2 + 1 = 2*5*13*41*661 -> 41 (end of the cycle because 41^2 + 1 = 2*29^2 -> 29 is already in the sequence).
%p A182782b := proc(n)
%p local bcyc,pfs ,b2;
%p bcyc := [n] ;
%p while true do
%p b2 := op(-1,bcyc) ;
%p pfs := sort(convert(numtheory[factorset](1+b2^2),list)) ;
%p endcy := true;
%p for f in pfs do
%p if not member(f,bcyc) then
%p endcy := false;
%p bcyc := [op(bcyc),f] ;
%p break;
%p end if;
%p end do:
%p if endcy then
%p return bcyc ;
%p end if;
%p end do;
%p end proc:
%p A182782 := proc(n)
%p nops(A182782b(n))-1 ;
%p end proc: # _R. J. Mathar_, Feb 06 2011
%K nonn
%O 1,1
%A _Michel Lagneau_, Feb 01 2011
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