

A030296


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


5



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
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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. 123, Spring 2001.


LINKS



FORMULA



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[n1]1; p2 = NextPrime[p1], True, p1 = p2; p2 = NextPrime[p1], If[ p2p11 >= n, Return[p1+1]]]; a[1] = 4; Table[a[n], {n, 1, 43}] (* JeanFrançois 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



KEYWORD

nonn,nice


AUTHOR



STATUS

approved



