A182782 - OEIS

%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