A114187 Difference between first semiprime >= n! and n!. Least k such that n!+k is semiprime.

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, 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, 118, 37, 1, 61

Offset: 0

Author

Jonathan Vos Post, Feb 04 2006

Keywords

• easy
• nonn

Comments

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

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

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.

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.

Essentially the same as A085747. - Georg Fischer, Oct 07 2018

Examples

			a(0) = a(1) = 3 because 0! + 3 = 1! + 3 = 4 = 2^2 is semiprime (the only even example).
a(2) = 2 because 2! + 2 = 2 + 2 = 4 = 2^2 is semiprime.
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).
a(4) = 1 because 4! + 1 = 24 + 1 = 25 = 5^2 is semiprime.
a(5) = 1 because 5! + 1 = 120 + 1 = 121 = 11^2 is semiprime.
a(6) = 1 because 6! + 1 = 720 + 1 = 721 = 7 * 103 is semiprime.
a(7) = 1 because 7! + 1 = 5040 + 1 = 5041 = 71^2 is semiprime.
a(8) = 1 because 8! + 1 = 40320 + 1 = 40321 = 61 * 661 is semiprime.
a(9) = 5 because 9! + 5 = 362880 + 1 = 362885 = 5 * 72577 is semiprime.
a(10) = 1 because 10! + 1 = 3628800 + 1 = 3628801 = 11 * 329891 is semiprime.
		

Crossrefs

Cf. A000142, A001358, A098147.

Formula

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.

Extensions

Data corrected by Giovanni Resta, Jun 14 2016