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).

16, 72, 540, 6300, 92400, 1681680, 36756720, 921470550, 27886608750, 970453984500, 37905932634570

Offset: 1

Author

Michael De Vlieger, Jun 22 2023

Keywords

• nonn
• hard
• more

Comments

Let R = r(P(n)) = A010846(A002110(n)) = A363061(n).

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.

Let T(n) = A002110(n)*S(n). Note that S(1) = T(1) since omega(A002110(1)) = 1.

Let S(n,i) be the i-th term in S(n).

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).

Examples

			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}.
a(2) = 72 = A003586(18) since r(72) = r(30) = 18. 72 is the 8th term in A003586 that is not in A000961.
a(3) = 540 since r(540) = 69 which exceeds r(210) = 68.
a(4) = 6300 since r(6300) = 290 which exceeds r(2310) = 283, etc.
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).
   n p(n)        P(n+1)          a(n)  r(P(n))  r(a(n))   j    m
  --------------------------------------------------------------
   1   2             6            16        5        5    4    8
   2   3            30            72       18       18    8   12
   3   5           210           540       68       69   13   18
   4   7          2310          6300      283      290   22   30
   5  11         30030         92400     1161     1165   29   40
   6  13        510510       1681680     4843     4848   42   56
   7  17       9699690      36756720    19985    19994   53   72
   8  19     223092870     921470550    83074    83435   68   95
   9  23    6469693230   27886608750   349670   351047   89  125
  10  29  200560490130  970453984500  1456458  1457926  107  150
		

Crossrefs

Cf. A000079, A000961, A002110, A002473, A003586, A007947, A010846, A051037, A051038, A080197, A080681, A080682, A080683, A162306, A363061.

Programs

Mathematica

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}]

Formula

a(n) >= A363061(n).