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

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

Offset: 1

Author

Mohamed Bouhamida, Aug 22 2009

Keywords

• nonn
• more

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

Examples

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

Crossrefs

Cf. A001358, A000045, A140362, A164643.

Programs

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

Extensions

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

a(7) from Max Alekseyev, Aug 27 2009