A114187 - OEIS

%I #13 Oct 08 2018 10:59:45

%S 3,3,2,0,1,1,1,1,1,5,1,3,17,1,1,7,2,3,23,1,1,11,29,3,1,1,1,37,1,41,2,

%T 19,11,11,1,7,3,41,1,13,127,47,59,2,37,5,37,59,1,2,73,59,79,73,1,1,61,

%U 118,37,1,61

%N Difference between first semiprime >= n! and n!. Least k such that n!+k is semiprime.

%C a(n) = 1 when n!+1 is a factorial prime.

%C A098147 is difference between first odd semiprime > 10^n and 10^n.

%C In this sequence, does 1 occur infinitely often (next with n = 71, 75)? If not 0 (for n=3) or 1, a(n) = k must be a prime other than 5.

%C Does every odd prime but 5 occur? Some of these take longer to factor, when both prime factors are large, such as n = 37, 38, 42, 47, 50, 54.

%C Essentially the same as A085747. - _Georg Fischer_, Oct 07 2018

%F a(n) = minimum integer k such that n! + k is an element of A001358. a(n) = minimum integer k such that A000142(n) + k is an element of A001358.

%e a(0) = a(1) = 3 because 0! + 3 = 1! + 3 = 4 = 2^2 is semiprime (the only even example).

%e a(2) = 2 because 2! + 2 = 2 + 2 = 4 = 2^2 is semiprime.

%e a(3) = 0 because 3! + 0 = 6 = 2*3 is semiprime (6+3=9=3^2 so this term would be 3 if we required nonzero values).

%e a(4) = 1 because 4! + 1 = 24 + 1 = 25 = 5^2 is semiprime.

%e a(5) = 1 because 5! + 1 = 120 + 1 = 121 = 11^2 is semiprime.

%e a(6) = 1 because 6! + 1 = 720 + 1 = 721 = 7 * 103 is semiprime.

%e a(7) = 1 because 7! + 1 = 5040 + 1 = 5041 = 71^2 is semiprime.

%e a(8) = 1 because 8! + 1 = 40320 + 1 = 40321 = 61 * 661 is semiprime.

%e a(9) = 5 because 9! + 5 = 362880 + 1 = 362885 = 5 * 72577 is semiprime.

%e a(10) = 1 because 10! + 1 = 3628800 + 1 = 3628801 = 11 * 329891 is semiprime.

%Y Cf. A000142, A001358, A098147.

%K easy,nonn

%O 0,1

%A _Jonathan Vos Post_, Feb 04 2006

%E Data corrected by _Giovanni Resta_, Jun 14 2016