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The nonprimes n!+2 ... n!+n are the a(n)-th string of n-1 prime-free consecutive terms, the first such one being the string of composite numbers A000230(k)+1 through A001632(k)-1 when n=2k, or through A001632(k)-2 when n=2k-1.
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%I #11 Jan 11 2023 05:07:30

%S 0,1,1,6,27,208,1755,16363,161685,1736749,20022517,250566242,

%T 3359504253

%N The nonprimes n!+2 ... n!+n are the a(n)-th string of n-1 prime-free consecutive terms, the first such one being the string of composite numbers A000230(k)+1 through A001632(k)-1 when n=2k, or through A001632(k)-2 when n=2k-1.

%C The sequence grows rapidly, like the factorial function.

%e The prime-free sequence 4! + 2 through 4! + 4, i.e., {26, 27, 28}, ranks as the a(4) = 6th triple of consecutive composite numbers, as it comes after {8, 9, 10}, {14, 15, 16}, {20, 21, 22}, {24, 25, 26}, {25, 26, 27}.

%t Do[ c = 0; a = Table[0, {n - 1} ]; k = 2; While[ k < n! + n + 1, a = Delete[a, 1]; a = Append[a, PrimeQ[k] ]; If[ Union[a] == {False}, c++ ]; k++ ]; Print[c], {n, 2, 12} ]

%Y Cf. A000230, A001632.

%K more,nonn,nice

%O 1,4

%A _Lekraj Beedassy_, May 10 2001

%E Corrected and extended by Larry Reeves (larryr(AT)acm.org), May 25 2001

%E More terms from _Robert G. Wilson v_, Aug 17 2001

%E a(13) from _Sean A. Irvine_, Jan 11 2023