A112888 Least semiprime of a cluster of just n semiprimes.

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

Offset: 1

Author

Robert G. Wilson v, Nov 30 2005

Keywords

• nonn
• fini
• full

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.

Examples

			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.
		

Crossrefs

Cf. A001358, A097824, A082919.

Programs

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 *)

Extensions

fini, full from Max Alekseyev, Feb 03 2010