A164643 Semiprimes pq with pq - 1 divisible by p + q.

6, 21, 301, 697, 1333, 1909, 2041, 3901, 24601, 26977, 96361, 130153, 163201, 250321, 275833, 296341, 389593, 486877, 495529, 542413, 808861, 1005421, 1005649, 1055833, 1063141, 1232053, 1284121, 1403221, 1618597, 1787917, 2287933, 2462881, 2488201, 2666437

Offset: 1

Author

Mohamed Bouhamida, Aug 19 2009

Keywords

• nonn

Comments

The first three terms are Syl(0)*Syl(1), Syl(1)*Syl(2) and Syl(2)*Syl(3). Syl means Sylvester's sequence, see A000058.

Products of two consecutive numbers p and q in Sylvester's sequence with primes p and q are in the sequence.

Let p and q be consecutive prime Sylvester numbers. Then: pq - 1 = p*(p^2 - p + 1) - 1 = p^3 - p^2 + p - 1 = (p^2 + 1)*(p - 1) = (p + p^2 - p + 1)*(p - 1) = (p + q)*(p - 1) it means that: (pq - 1) is divisible by (p + q). - Mohamed Bouhamida, Aug 21 2009

(p-k)*(q-k) = k^2 + 1 for some integer k, providing a fast way for finding appropriate p,q. - Max Alekseyev, Aug 26 2009

Links

• Donovan Johnson, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001358, A000058.

Programs

Maple

isA001358 := proc(n) RETURN ( numtheory[bigomega](n) =2 ) ; end:
isA164643 := proc(n) if isA001358(n) then p := op(1,op(1,ifactors(n)[2]) ) ; q := n/p ; if (p*q-1) mod (p+q) =0 then true; else false; fi; else false; fi; end:
for n from 4 to 3000000 do if isA164643(n) then print(n) ; fi; od: # R. J. Mathar, Aug 24 2009

Mathematica

dsQ[n_]:=Module[{prs=Transpose[FactorInteger[n]][[1]]},Divisible[n-1, Total[prs]]]; Select[Select[Range[2000000], PrimeOmega[#] ==2&], dsQ] (* Harvey P. Dale, Jun 15 2011 *)

Extensions

Extended by R. J. Mathar, Aug 24 2009

More terms from Max Alekseyev, Aug 26 2009