Consecutive smooth triplets

Question

Consecutive $n$-smooth triplets with no common factors are possible. The sequence 64, 120, 324, 2024, 17576, 248676, 314432, 6571774, 7496644, 116026274, 196512876 isn't in OEIS, but they do appear in Koninck's Those Fascinating Numbers. For each of these, $k-1, k, k+1$ all have a rather low maximum prime factor. For consecutive smooth pairs, Størmer's theorem can be used. That's how I verified these up to 97-smooth, but that took my current program a week. Values for 101 and 113 are unverified.

If the "no common factors" is dropped, the middle-odd $n$-smooth triples always seem to be higher than the middle-even $n$-smooth triples.

Can anyone extend, improve, or correct these results?

Answer 1

I found a better result for 113:

1129770949: 41 43 53 107 113
1129770950: 2 5 5 7 7 11 11 37 103
1129770951: 3 19 29 67 101 101

You can use a fairly simple sieve algorithm to find smallish solutions very quickly if you use a reasonable programming language. My program finds all primes less than the smoothness bound and computes $v_p = \lfloor 4000 \log_{10}(p) \rfloor$ for each such prime. If you are looking for solutions in the interval $[L, L+N]$, add up $v_p$ for each small prime divisor of each number in the interval. Dump out all consecutive triplets such that the sum of relevant $v_p$'s is larger than $\lfloor 4000 \log_{10}(L) \rfloor - 4000$. Postprocess the results manually to weed out any false positives.

EDIT: I left said sieve running while I was at work today. For the common-factors case, the following solution may also be interesting:

138982582998: 2 3 3 3 3 3 3 7 29 47 97 103
138982582999: 13 31 37 43 43 71 71
138982583000: 2 2 2 5 5 5 23 23 59 61 73