A136798 First term in a sequence of at least 3 consecutive composite integers.

8, 14, 20, 24, 32, 38, 44, 48, 54, 62, 68, 74, 80, 84, 90, 98, 104, 110, 114, 128, 132, 140, 152, 158, 164, 168, 174, 182, 194, 200, 212, 224, 230, 234, 242, 252, 258, 264, 272, 278, 284, 294, 308, 314, 318, 332, 338, 350, 354, 360, 368, 374, 380, 384, 390, 398

Offset

1,1

Comments

The meaning of "first" is that the run of composites is started with this term, that is, it is the one after a prime.

The number of terms in any run of composites is odd, because the difference between the relevant consecutive primes is even.

Composite numbers m such that m+1 is also composite, but m-1 is not. - Reinhard Zumkeller, Aug 04 2015

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Carlos Rivera, Puzzle 430, Grimm's Conjecture, Prime puzzles and problems connection.

Formula

a(n) = A049591(n)+1. - R. J. Mathar, Jan 23 2008

A010051(a(n)-1) * (1 - A010051(a(n)) - A010051(a(n)+1)) = 1. - Reinhard Zumkeller, Aug 04 2015

Example

a(1)=8 because 8 is the first term in a sequential run of 3 composites, 8,9,10

Mathematica

Prime/@Flatten[Position[Differences[Prime[Range[80]]], _?(#>2&)]]+1 (* Harvey P. Dale, Jun 19 2013 *)

Prog

(Haskell)
import Data.List (elemIndices)
a136798 n = a136798_list !! (n-1)
a136798_list = tail $ map (+ 1) $ elemIndices 1 $
   zipWith (*) (0 : a010051_list) $ map (1 -) $ tail a010051_list
-- Reinhard Zumkeller, Aug 04 2015

Crossrefs

Cf. A136799, A136800, A136801.

Cf. A049591, A010051.

a(n) = 2 * A104280(n).

Sequence in context: A309355 A374223 A063288 * A172182 A091575 A091572

Adjacent sequences: A136795 A136796 A136797 * A136799 A136800 A136801

Keyword

easy,nonn

Author

Enoch Haga, Jan 21 2008

Extensions

Edited by R. J. Mathar, May 27 2009

Status

approved