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.
%I A363794 #13 Jul 08 2024 08:51:06
%S A363794 16,72,540,6300,92400,1681680,36756720,921470550,27886608750,
%T A363794 970453984500,37905932634570
%N A363794 a(n) = smallest prime(n)-smooth number k such that r(k) >= r(P(n+1)), where r(n) = A010846(n) and P(n) = A002110(n).
%C A363794 Let R = r(P(n)) = A010846(A002110(n)) = A363061(n).
%C A363794 Let S(n) be the sorted tensor product of prime power ranges {p(i)^e : i<=n, e>=0}, e.g., S(1) = A000079, S(2) = A003586, S(3) = A051037, etc.
%C A363794 Let T(n) = A002110(n)*S(n). Note that S(1) = T(1) since omega(A002110(1)) = 1.
%C A363794 Let S(n,i) be the i-th term in S(n).
%C A363794 Then a(n) is the smallest S(n,i), i >= R, such that S(n,i) is also in T. Equivalently, a(n) is the smallest S(n,i), i >= R, such that rad(S(n,i)) = A002110(n), where rad(n) = A007947(n).
%F A363794 a(n) >= A363061(n).
%e A363794 a(1) = 16 since r(2^4) = 5 and r(6) = 5; numbers in row 16 of A162306 are its divisors {1, 2, 4, 8, 16}, while row 6 of A162306 is {1, 2, 3, 4, 6}.
%e A363794 a(2) = 72 = A003586(18) since r(72) = r(30) = 18. 72 is the 8th term in A003586 that is not in A000961.
%e A363794 a(3) = 540 since r(540) = 69 which exceeds r(210) = 68.
%e A363794 a(4) = 6300 since r(6300) = 290 which exceeds r(2310) = 283, etc.
%e A363794 Table showing the relationship of a(n) to r(P(n)) = A363061(n), with p(n) = prime(n), P(n+1) = A002110(n+1), r(a(n)) = A010846(a(n)), and j the index such that S(r(a(n))) = T(j) = a(n). a(n) = m*P(n).
%e A363794 n p(n) P(n+1) a(n) r(P(n)) r(a(n)) j m
%e A363794 --------------------------------------------------------------
%e A363794 1 2 6 16 5 5 4 8
%e A363794 2 3 30 72 18 18 8 12
%e A363794 3 5 210 540 68 69 13 18
%e A363794 4 7 2310 6300 283 290 22 30
%e A363794 5 11 30030 92400 1161 1165 29 40
%e A363794 6 13 510510 1681680 4843 4848 42 56
%e A363794 7 17 9699690 36756720 19985 19994 53 72
%e A363794 8 19 223092870 921470550 83074 83435 68 95
%e A363794 9 23 6469693230 27886608750 349670 351047 89 125
%e A363794 10 29 200560490130 970453984500 1456458 1457926 107 150
%t A363794 nn = 6; rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; f[x_] := FactorInteger[x][[-1, 1]]; S = Array[Product[Prime[i], {i, #}] &, nn + 1]; Table[Set[{p, q}, Prime[n + {0, 1}]]; r = Count[Range[S[[n + 1]]], _?(f[#] <= q &)]; c = k = 1; While[Or[c < r, rad[k] != S[[n]]], If[f[k] <= p, c++]; k++]; k, {n, nn}]
%Y A363794 Cf. A000079, A000961, A002110, A002473, A003586, A007947, A010846, A051037, A051038, A080197, A080681, A080682, A080683, A162306, A363061.
%K A363794 nonn,hard,more
%O A363794 1,1
%A A363794 _Michael De Vlieger_, Jun 22 2023