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

%I A030296 #35 Jul 08 2025 19:32:53
%S A030296 4,8,8,24,24,90,90,114,114,114,114,114,114,524,524,524,524,888,888,
%T A030296 1130,1130,1328,1328,1328,1328,1328,1328,1328,1328,1328,1328,1328,
%U A030296 1328,9552,9552,15684,15684,15684,15684,15684,15684,15684,15684,19610,19610,19610
%N A030296 Smallest start for a run of at least n composite numbers.
%C A030296 a(n) is even, since a(n)-1 is a prime > 2, by the minimality of a(n). - _Jonathan Sondow_, May 31 2014
%C A030296 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
%D A030296 Amarnath Murthy, Some more conjectures on primes and divisors, Smarandache Notions Journal, Vol. 12, No. 1-2-3, Spring 2001.
%H A030296 Donovan Johnson, <a href="/A030296/b030296.txt">Table of n, a(n) for n = 1..1475</a> (terms < 4*10^18)
%H A030296 Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html">First occurrence prime gaps</a> [For local copy see A000101]
%H A030296 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeGaps.html">Prime Gaps</a>
%F A030296 a(n) = A104138(n) + 1. - _Jonathan Sondow_, May 31 2014
%e A030296 a(5) = 24 as 24 is the first of the five consecutive composite numbers 24, 25, 26, 27, 28.
%t A030296 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 *)
%t A030296 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 *)
%Y A030296 Cf. A008950, A008995, A008996, A000101, A002386, A104138.
%K A030296 nonn,nice
%O A030296 1,1
%A A030296 _Eric W. Weisstein_