A362044 a(n) = largest k such that k < m^2 and rad(k) | m, where rad(k) = A007947(k) and m = A120944(n).

32, 80, 128, 135, 343, 352, 512, 864, 891, 1088, 875, 1216, 1053, 1728, 2048, 2187, 1375, 2187, 2048, 2048, 3125, 4224, 2187, 4802, 4736, 3773, 5832, 5248, 4913, 5504, 7047, 4459, 7533, 8192, 6859, 10368, 10935, 8192, 11264, 8991, 12312, 12167, 8192, 5831, 8192, 9963, 10449, 16640, 16807, 17152, 18432

Offset: 1

Author

Michael De Vlieger, Apr 05 2023

Keywords

• nonn

Comments

The largest k such that k < p^2 such that p is prime and rad(k) | p is p itself.

Examples

			a(1) = 32 since m = 6 and the largest k < m^2 such that rad(k) | 6 is 32. This is to say, the number that precedes 6^2 in A003586 is 32.
a(2) = 80 since m = 10 and the largest k < m^2 such that rad(k) | 10 is 80. This is to say, the number that precedes 10^2 in A003592 is 80.
Table of n = 1..12, m = A120944(n), a(n), and m^2.
   n    m    a(n)   m^2
  ---------------------
   1    6     32     36
   2   10     80    100
   3   14    128    196
   4   15    135    225
   5   21    343    441
   6   22    352    484
   7   26    512    676
   8   30    864    900
   9   33    891   1089
  10   34   1088   1156
  11   35    875   1225
  12   38   1216   1444
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Scatterplot of a(n), m^2, and b(n), n = 1..2^14, where b(n) = A362045(n) is shown in red, m^2 in black, and a(n) in blue.

Crossrefs

Cf. A007947, A120944, A289280, A362045.

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[#]] &]}]