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

%I A164643 #13 Feb 15 2020 10:52:27
%S A164643 6,21,301,697,1333,1909,2041,3901,24601,26977,96361,130153,163201,
%T A164643 250321,275833,296341,389593,486877,495529,542413,808861,1005421,
%U A164643 1005649,1055833,1063141,1232053,1284121,1403221,1618597,1787917,2287933,2462881,2488201,2666437
%N A164643 Semiprimes pq with pq - 1 divisible by p + q.
%C A164643 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.
%C A164643 Products of two consecutive numbers p and q in Sylvester's sequence with primes p and q are in the sequence.
%C A164643 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
%C A164643 (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
%H A164643 Donovan Johnson, <a href="/A164643/b164643.txt">Table of n, a(n) for n = 1..1000</a>
%p A164643 isA001358 := proc(n) RETURN ( numtheory[bigomega](n) =2 ) ; end:
%p A164643 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:
%p A164643 for n from 4 to 3000000 do if isA164643(n) then print(n) ; fi; od: # _R. J. Mathar_, Aug 24 2009
%t A164643 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 *)
%Y A164643 Cf. A001358, A000058.
%K A164643 nonn
%O A164643 1,1
%A A164643 _Mohamed Bouhamida_, Aug 19 2009
%E A164643 Extended by _R. J. Mathar_, Aug 24 2009
%E A164643 More terms from _Max Alekseyev_, Aug 26 2009