A376302 Smallest powerful m in row n of A064364, or -1 if none exist.

1, -1, -1, 4, -1, 8, -1, 16, 27, 25, -1, 64, 108, 49, 125, 200, 432, 196, 500, 392, 343, 121, 1323, 1225, 1372, 169, 2744, 968, 5488, 676, 3267, 1352, 1331, 289, 4563, 4225, 5324, 361, 2197, 2312, 21125, 1444, 7803, 2888, 17576, 529, 9747, 9025, 36125, 2116, 4913

Offset: 1

Author

Michael De Vlieger, Sep 19 2024

Keywords

• sign

Comments

Powerful numbers here refers to A001694.

Examples

			First 13 rows of A064364 indicating the smallest powerful number in each row with brackets and other powerful numbers in parentheses. Rows 2, 3, 5, 7, and 11 do not have powerful numbers.
   n   Row n of A064364
  -------------------------------------------------
   1:  [1]
   2:   2
   3:   3
   4:  [4]
   5:   5    6
   6:  [8]  (9)
   7:   7   10   12
   8:  15  [16]  18
   9:  14   20   24 [27]
  10:  21  [25]  30   32  (36)
  11:  11   28   40   45   48   54
  12:  35   42   50   60  [64]  72  (81)
  13:  13   22   56   63   75   80   90   96  [108]
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..10766 (i.e., a(n) <= 2^40), ignoring negative values, showing perfect prime powers in red, and other powerful numbers in dark blue.

Crossrefs

Cf. A001414, A001694, A064364, A363266 (largest squarefree m in row n of A064364).

Programs

Mathematica

s = With[{nn = 40000}, Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]},
  {a, Sqrt[nn/b^3]}]];
Insert[#, -1, Map[List, {2, 2, 3, 4, 7}]] &@
  s[[Values[#[[7 ;; 7 + LengthWhile[Differences@
  Keys[#][[7 ;; -1]], # == 1 &] ]] ][[All, 1]] ]] &@
  KeySort@ PositionIndex@
  Map[Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]] &, s]