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
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