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

%I A376302 #5 Sep 20 2024 06:01:35
%S A376302 1,-1,-1,4,-1,8,-1,16,27,25,-1,64,108,49,125,200,432,196,500,392,343,
%T A376302 121,1323,1225,1372,169,2744,968,5488,676,3267,1352,1331,289,4563,
%U A376302 4225,5324,361,2197,2312,21125,1444,7803,2888,17576,529,9747,9025,36125,2116,4913
%N A376302 Smallest powerful m in row n of A064364, or -1 if none exist.
%C A376302 Powerful numbers here refers to A001694.
%H A376302 Michael De Vlieger, <a href="/A376302/b376302.txt">Table of n, a(n) for n = 1..10000</a>
%H A376302 Michael De Vlieger, <a href="/A376302/a376302.png">Log log scatterplot of a(n)</a>, 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.
%e A376302 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.
%e A376302    n   Row n of A064364
%e A376302   -------------------------------------------------
%e A376302    1:  [1]
%e A376302    2:   2
%e A376302    3:   3
%e A376302    4:  [4]
%e A376302    5:   5    6
%e A376302    6:  [8]  (9)
%e A376302    7:   7   10   12
%e A376302    8:  15  [16]  18
%e A376302    9:  14   20   24 [27]
%e A376302   10:  21  [25]  30   32  (36)
%e A376302   11:  11   28   40   45   48   54
%e A376302   12:  35   42   50   60  [64]  72  (81)
%e A376302   13:  13   22   56   63   75   80   90   96  [108]
%t A376302 s = With[{nn = 40000}, Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]},
%t A376302   {a, Sqrt[nn/b^3]}]];
%t A376302 Insert[#, -1, Map[List, {2, 2, 3, 4, 7}]] &@
%t A376302   s[[Values[#[[7 ;; 7 + LengthWhile[Differences@
%t A376302   Keys[#][[7 ;; -1]], # == 1 &] ]] ][[All, 1]] ]] &@
%t A376302   KeySort@ PositionIndex@
%t A376302   Map[Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]] &, s]
%Y A376302 Cf. A001414, A001694, A064364, A363266 (largest squarefree m in row n of A064364).
%K A376302 sign
%O A376302 1,4
%A A376302 _Michael De Vlieger_, Sep 19 2024