This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(1) = 2 because 30 = 2 * 3 * 5 has third largest prime factor 2. a(2) = 2 because 42 = 2 * 3 * 7 has third largest prime factor 2. a(3) = 2 because 60 = 2 * 2 * 3 * 5 has both third and fourth largest prime factor 2. a(8) = 3 because 90 = 2 * 3 * 3 * 5 has both second and third largest prime factor 3.
b:= proc(n) option remember; local k;
if n=1 then 30
else for k from b(n-1)+1 while
nops(ifactors(k)[2])<3 do od;
k
fi
end:
a:= n-> sort(map(x-> x[1]$x[2], ifactors(b(n))[2]))[-3]:
seq(a(n), n=1..120);
b[n_] := b[n] = Module[{k}, If[n==1, 30, For[k = b[n-1]+1, PrimeNu[k] < 3, k++]; k]];
a[n_] := (Table[#[[1]], {#[[2]]}]& /@ FactorInteger[b[n]] // Flatten // Sort)[[-3]];
Array[a, 120] (* Jean-François Alcover, Nov 28 2020, after Alois P. Heinz *)
The number of distinct prime factors of the numbers 15, 14, 13, 12, 11, 10 are respectively 2, 2, 1, 2, 1, 2 and 2*2*1*2*1*2 = 16, hence 16 is a term.
om[n_] := om[n] = PrimeNu[n]; q[n_] := Module[{m = n, k = n - 1}, While[k > 1 && Divisible[m, om[k]], m /= om[k]; k--]; m == 1]; Select[Range[2, 10^6], q] (* Amiram Eldar, Nov 02 2021 *)
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