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 A164643 Semiprimes pq with pq - 1 divisible by p + q. 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 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 *) CROSSREFS Cf. A001358, A000058. Sequence in context: A244299 A143049 A213680 * A190275 A261844 A007594 Adjacent sequences:  A164640 A164641 A164642 * A164644 A164645 A164646 KEYWORD nonn AUTHOR Mohamed Bouhamida, Aug 19 2009 EXTENSIONS Extended by R. J. Mathar, Aug 24 2009 More terms from Max Alekseyev, Aug 26 2009 STATUS approved

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Last modified August 12 23:52 EDT 2022. Contains 356077 sequences. (Running on oeis4.)