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

Author

Enoch Haga, Jan 21 2008

Keywords

• easy
• nonn

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

Examples

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

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.

Crossrefs

Cf. A136799, A136800, A136801.

Cf. A049591, A010051.

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

Programs

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

Mathematica

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

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

Extensions

Edited by R. J. Mathar, May 27 2009