A030296 Smallest start for a run of at least n composite numbers.

4, 8, 8, 24, 24, 90, 90, 114, 114, 114, 114, 114, 114, 524, 524, 524, 524, 888, 888, 1130, 1130, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 9552, 9552, 15684, 15684, 15684, 15684, 15684, 15684, 15684, 15684, 19610, 19610, 19610

Offset

1,1

Comments

a(n) is even, since a(n)-1 is a prime > 2, by the minimality of a(n). - Jonathan Sondow, May 31 2014

Except for a(1), records occur at even values of n, and each term appears an even number of times consecutively. (Proof. A maximal run of composites must begin and end at even numbers.) - Jonathan Sondow, May 31 2014

References

Amarnath Murthy, Some more conjectures on primes and divisors, Smarandache Notions Journal, Vol. 12, No. 1-2-3, Spring 2001.

Links

Donovan Johnson, Table of n, a(n) for n = 1..1475 (terms < 4*10^18)

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]

Eric Weisstein's World of Mathematics, Prime Gaps

Formula

a(n) = A104138(n) + 1. - Jonathan Sondow, May 31 2014

Example

a(5) = 24 as 24 is the first of the five consecutive composite numbers 24, 25, 26, 27, 28.

Mathematica

a[n_] := a[n] = For[p1 = a[n-1]-1; p2 = NextPrime[p1], True, p1 = p2; p2 = NextPrime[p1], If[ p2-p1-1 >= n, Return[p1+1]]]; a[1] = 4; Table[a[n], {n, 1, 43}] (* Jean-François Alcover, May 24 2012 *)
Module[{nn=20000, cmps}, cmps=Table[If[CompositeQ[n], 1, 0], {n, nn}]; Table[ SequencePosition[ cmps, PadRight[{}, k, 1], 1][[1, 1]], {k, 50}]] (* Harvey P. Dale, Jan 01 2022 *)

Crossrefs

Cf. A008950, A008995, A008996, A000101, A002386, A104138.

Sequence in context: A141800 A019197 A066106 * A160171 A195752 A095806

Adjacent sequences: A030293 A030294 A030295 * A030297 A030298 A030299

Keyword

nonn,nice

Author

Eric W. Weisstein.

Status

approved