1,3

Numbers n such that a(n) = 0 are listed in A125212(n) = {1,2,10,16,28,34,36,40,46,50,51,52,56,57,58,64,66,70,76,78,82,86,87,88,92,93,94,96,100,...} Numbers n such that no prime exists of the form k! - n. Note the triplets of consecutive zeros in a(n) for n = {{50,51,52}, {56,57,58}, {86,87,88}, {92,93,94}, ...}. Most zeros in a(n) have even indices. The middle index of most consecutive zero triplets is odd and is a multiple of 3. Numbers n such that no prime exists of the form (k! - 3n - 1), (k! - 3n), (k! - 3n + 1) are listed in A125213(n) = {17,19,29,31,45,49,57,59,63,69,73,79,83,85,87,89,97,99,...}. The first pair of odd middle indices of zero triplets that are not divisible by 3 is n = 325 and n = 329. They belong to the first septuplet of consecutive zeros in a(n): a(324)-a(330) = 0.

a(4) = 3 because there are 3 positive and negative primes of the form k! - 4:

1! - 4 = -3, 2! - 4 = -2, 3! - 4 = 2.

k! - 4 is composite for all k>3 because it is divisible by 4.

Table[Length[Select[Range[n], PrimeQ[ #!-n]&]], {n, 1, 300}]

Cf. A125162 = number of primes of the form k! + n. Cf. A125163 = numbers n such that no prime exists of the form k! + n. Cf. A125164 = numbers n such that no prime exists of the form (k! + 3n - 1), (k! + 3n), (k! + 3n + 1). Cf. A125212 = numbers n such that no positive and no negative prime exists of the form k! - n. Cf. A125213 = numbers n such that no positive and no negative prime exists of the form (k! - 3n - 1), (k! - 3n), (k! - 3n + 1).

Sequence in context: A134819 A135267 A118105 this_sequence A139367 A117648 A037222

Adjacent sequences: A125208 A125209 A125210 this_sequence A125212 A125213 A125214

nonn

Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 23 2006