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 b-sequence.

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

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 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

Sequence in context: A070561 A059941 A086100 * A217789 A347765 A004283

Adjacent sequences: A182779 A182780 A182781 * A182783 A182784 A182785

Keyword

nonn

Author

Michel Lagneau, Feb 01 2011

Status

approved