OFFSET
1,1
COMMENTS
Clusters are sets composed of odd numbers.
If we include even numbers then the sequence would start 4,9,33 and terminates because in any group of four consecutive numbers greater than 4, 4 is a divisor to at least one member leaving a quotient greater than 1.
Any set of 9 consecutive odd numbers contain a multiple of 9, which not semiprime (unless it is equal to 9). Hence there are no 9 consecutive odd semiprimes.
EXAMPLE
a(8)=8129 because 8129=11*739, 8131=47*173, 8133=3*2711, 8135=5*1627, 8137=79*103, 8139=3*2713, 8141=7*1163, 8143=17*479.
MATHEMATICA
spQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; f[n_] := Block[{k = 1}, While[ s[[k]] + 2n != s[[k + n]] || s[[k]] + 2n + 2 == s[[k + n + 1]], k++ ]; s[[k]]]; s = {}; Do[ If[ spQ[n], AppendTo[s, n]], {n, 9, 7*10^6, 2}]; Table[ f[n], {n, 0, 7}]
Join[{9}, Module[{osps=Select[Range[9, 10001, 2], PrimeOmega[#]==2&]}, #[[2]]& /@ Table[ SelectFirst[Partition[osps, n+2, 1], Union[ Differences[ Rest[ Most[#]]]]=={2}&&Last[#]-#[[-2]]!=2&&#[[2]]-#[[1]]!=2&], {n, 2, 8}]]] (* Harvey P. Dale, Jun 01 2016 *)
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Robert G. Wilson v, Nov 30 2005
EXTENSIONS
fini, full from Max Alekseyev, Feb 03 2010
STATUS
approved