A362045 a(n) = smallest k such that k > m^2 and rad(k) | m, where rad(k) = A007947(k) and m = A120944(n).

48, 125, 224, 243, 567, 512, 832, 960, 1331, 2048, 1715, 2048, 2187, 1792, 2944, 4131, 3125, 4617, 3712, 3968, 8125, 4374, 5589, 5000, 8192, 9317, 6144, 8192, 10625, 8192, 19683, 15379, 19683, 12032, 11875, 11016, 11907, 13568, 12500, 19683, 13122, 14375, 15104, 16807, 15616, 19683, 19683, 17576, 45619

Offset: 1

Author

Michael De Vlieger, Apr 05 2023

Keywords

• nonn

Comments

The smallest k such that k > p^2 such that p is prime and rad(k) | p is p^3.

Examples

			a(1) = 48 since m = 6 and the smallest k > m^2 such that rad(k) | 6 is 48. This is to say, the number that follows 6^2 in A003586 is 48.
a(2) = 80 since m = 10 and the smallest k > m^2 such that rad(k) | 10 is 125. This is to say, the number that precedes 10^2 in A003592 is 125.
Table of n = 1..12, m = A120944(n), m^2, and a(n).
   n    m    m^2   a(n)
  ---------------------
   1    6     36     48
   2   10    100    125
   3   14    196    224
   4   15    225    243
   5   21    441    567
   6   22    484    512
   7   26    676    832
   8   30    900    960
   9   33   1089   1331
  10   34   1156   2048
  11   35   1225   1715
  12   38   1444   2048
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..32768
• Michael De Vlieger, Scatterplot of a(n), m^2, and b(n), n = 1..2^14, where b(n) = A362044(n) is shown in blue, m^2 in black, and a(n) in red.

Crossrefs

Cf. A007947, A120944, A289280, A362044.

Programs

Mathematica

Table[m = k^2 + 1; While[! Divisible[k, Times @@ FactorInteger[m][[All, 1]]], m++]; m, {k, Select[Range[6, 133], And[CompositeQ[#], SquareFreeQ[#]] &]}]