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.
EXAMPLE
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:
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Feb 01 2011
STATUS
approved