

A182782


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 bsequence.


1



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, 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, 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
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OFFSET

1,1


COMMENTS

The number of terms of each sequence b(n) is finite.
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.


LINKS

Table of n, a(n) for n=1..109.


EXAMPLE

a(1) = 11 counts the iterations in the bchain 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:
1^2 + 1 = 2;
2^2 + 1 = 5;
5^2 + 1 = 2*13 > 13 because 2 is already in the sequence;
13^2 + 1 = 2*5*17 > 17 because 2 and 5 are already in the sequence;
17^2 + 1 = 2*5*29 > 29;
29^2 + 1 = 2*421 > 421;
421^2 + 1 = 2*13*17*401 > 401;
401^2 + 1 = 2*37*41*53 > 37;
37^2 + 1 = 2*5*137 > 137;
137^2 + 1 = 2*5*1877 > 1877;
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).


MAPLE

A182782b := proc(n)
local bcyc, pfs , b2;
bcyc := [n] ;
while true do
b2 := op(1, bcyc) ;
pfs := sort(convert(numtheory[factorset](1+b2^2), list)) ;
endcy := true;
for f in pfs do
if not member(f, bcyc) then
endcy := false;
bcyc := [op(bcyc), f] ;
break;
end if;
end do:
if endcy then
return bcyc ;
end if;
end do;
end proc:
A182782 := proc(n)
nops(A182782b(n))1 ;
end proc: # R. J. Mathar, Feb 06 2011


CROSSREFS

Sequence in context: A070561 A059941 A086100 * A217789 A004283 A106421
Adjacent sequences: A182779 A182780 A182781 * A182783 A182784 A182785


KEYWORD

nonn


AUTHOR

Michel Lagneau, Feb 01 2011


STATUS

approved



