Number of composite numbers between S000564(n) and the next prime number.

1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 9, 9, 9, 9, 11, 9, 9, 11, 9, 11, 13, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 17, 15, 15, 17, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 21, 19, 19, 21, 21, 21, 21, 21, 21, 23, 21, 23

2

It appears that each positive odd number occurs here about four times.

T. D. Noe, Plot of 499 terms

T. D. Noe, Table of 499 terms

Eric W. Weisstein, MathWorld: Primes Gaps

(Mma) nn = 10000; t = Join[{0}, Table[Total[Transpose[FactorInteger[n]][[2]]], {n, 2, Prime[nn]}]]; t2 = Table[Total[t[[Range[Prime[n] + 1, Prime[n + 1] - 1]]]], {n, nn - 1}]; t3 = Sort[t2]; missing = Complement[Range[t3[[-1]]], t3][[2]]; t4 = Table[Prime[Position[t2, n, 1, 1][[1, 1]]], {n, 2, missing - 1}]; Table[NextPrime[n] - n - 1, {n, t4}]

Cf. S000564.

nonn

T. D. Noe, Apr 06 2015