A164698 Semiprimes pq such that pq - 1 divides p^2 + q^2 + 2.

6, 21, 26, 51, 1157, 372101, 1288005205276048901

Offset

1,1

Comments

Semiprimes pq such that pq-1 divides (p+q)^2.

The third to fifth terms are Fib(3)*Fib(7), Fib(7)*Fib(11) and Fib(13)*Fib(17).

Products of two prime Fibonacci numbers F(k) and F(k+4) (see A001605 and A005478) are in the sequence.

6 and 26 are the only even terms. Odd terms contain products of pairs of consecutive terms from the following sequences: A005248, A001541, A033889, A033891. - Max Alekseyev, Aug 27 2009

Links

Table of n, a(n) for n=1..7.

Example

The semiprime 6 = 2*3 is in the sequence because 2*3 - 1 = 5 divides 2^2 + 3^2 + 2 = 15.

Maple

isA001358 := proc(n) RETURN ( numtheory[bigomega](n) =2 ) ; end:
isA164698 := proc(n) if isA001358(n) then p := op(1, op(1, ifactors(n)[2]) ) ; q := n/p ; if (p^2+q^2+2) mod (p*q-1) = 0 then true; else false; fi; else false; fi; end:
for n from 4 to 3000000 do if isA164698(n) then print(n, ifactors(n)) ; fi; od: # R. J. Mathar, Aug 24 2009

Crossrefs

Cf. A001358, A000045, A140362, A164643.

Sequence in context: A327865 A369969 A229498 * A020880 A046467 A132184

Adjacent sequences: A164695 A164696 A164697 * A164699 A164700 A164701

Keyword

nonn,more

Author

Mohamed Bouhamida, Aug 22 2009

Extensions

Missing values added by R. J. Mathar, Aug 24 2009

a(7) from Max Alekseyev, Aug 27 2009

Status

approved