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A164643
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Semiprimes pq with pq - 1 divisible by p + q.
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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). [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Aug 21 2009]
(p-k)*(q-k) = k^2 + 1 for some integer k, providing a fast way for finding appropriate p,q [From Max Alekseyev (maxale(AT)gmail.com), Aug 26 2009]
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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
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MATHEMATICA
| dsQ[n_]:=Module[{prs=Transpose[FactorInteger[n]][[1]]}, Divisible[n-1, Total[prs]]]; Select[Select[Range[2000000], PrimeOmega[#] ==2&], dsQ] (* From Harvey P. Dale, June 15 2011 *)
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CROSSREFS
| Cf. A001358, A000058.
Sequence in context: A054366 A179768 A143049 * A190275 A007594 A181593
Adjacent sequences: A164640 A164641 A164642 * A164644 A164645 A164646
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KEYWORD
| nonn
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AUTHOR
| Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Aug 19 2009
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EXTENSIONS
| Extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 24 2009
More terms from Max Alekseyev (maxale(AT)gmail.com), Aug 26 2009
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