A112888 Least semiprime of a cluster of just n semiprimes.

9, 33, 91, 299, 213, 1383, 3091, 8129

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.

Links

Table of n, a(n) for n=1..8.

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

Cf. A001358, A097824, A082919.

Sequence in context: A020326 A201024 A228170 * A048479 A031880 A231765

Adjacent sequences: A112885 A112886 A112887 * A112889 A112890 A112891

Keyword

nonn,fini,full

Author

Robert G. Wilson v, Nov 30 2005

Extensions

fini, full from Max Alekseyev, Feb 03 2010

Status

approved