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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. 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 (list; graph; refs; listen; history; text; internal format)
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 A004283 A106421

Adjacent sequences:  A182779 A182780 A182781 * A182783 A182784 A182785

KEYWORD

nonn

AUTHOR

Michel Lagneau, Feb 01 2011

STATUS

approved

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Last modified October 16 13:32 EDT 2019. Contains 328093 sequences. (Running on oeis4.)