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A340803 Number of composite numbers between the largest noncomposite number <= prime(n)! and the smallest noncomposite number >= prime(n)!.

%I A340803 #18 Jan 29 2021 08:12:24
%S A340803 0,1,13,11,13,89,89,189,185,253,187,97,131,179,311,129,429,839,263,
%T A340803 1607,373,527,509,305,539,617,819,1687,565,423,2345,959,629,631,647,
%U A340803 1093,835,1937,491,1241,2171,1139,731,1423,2011,1817,1935,827,1247,911,2669
%N A340803 Number of composite numbers between the largest noncomposite number <= prime(n)! and the smallest noncomposite number >= prime(n)!.
%H A340803 Alois P. Heinz, <a href="/A340803/b340803.txt">Table of n, a(n) for n = 1..300</a>
%F A340803 a(n) = A340802(A000040(n)).
%e A340803 a(2) = 1: 6 because prime(2) = 3; 3! = 6 whose neighbors are noncomposites.
%e A340803 a(3) = 13: 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126.
%e A340803 a(4) = 11: 5040, 5041, 5042, 5043, 5044, 5045, 5046, 5047, 5048, 5049, 5050.
%p A340803 a:= n-> (f-> max(nextprime(f-1)-prevprime(f+1)-1, 0))(ithprime(n)!):
%p A340803 seq(a(n), n=1..52);
%t A340803 b[n_] := If[n < 3, 0, NextPrime[n!] - NextPrime[n!, -1] - 1];
%t A340803 a[n_] := b[Prime[n]];
%t A340803 Array[a, 52] (* _Jean-François Alcover_, Jan 29 2021 *)
%Y A340803 Cf. A000040, A002808, A008578, A046933, A340802.
%K A340803 nonn
%O A340803 1,3
%A A340803 _Alois P. Heinz_, Jan 21 2021