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%I #7 Dec 06 2024 11:10:14
%S 5,11,10,1,10,19,1,4,2,22,12,27,1,11,2,14,6,28,26,9,0,41,3,26,13,25,0,
%T 10,35,11,10,0,26,26,8,10,5,26,30,17,11,52,13,12,56,1,20,9,34,69,1,69,
%U 37,3,38,0,14,57,11,39,23,15,26,18,6,36,3,30,27,27,97
%N Number of k in A126706 between powerful numbers that are not prime powers.
%C Within the sequence S = A126706 of powerful numbers, we have numbers k that are powerful (in A286708) and numbers m that are not powerful (in A332785). This sequence is the number of k between m.
%H Michael De Vlieger, <a href="/A378700/b378700.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael De Vlieger, <a href="/A378700/a378700.png">Log log scatterplot of a(n)</a>, n = 1..11210, rendering a(n) = 0 instead as 1/2 for visibility.
%e We partition S = A126706 by numbers k in A286708 (in brackets) and derive the following irregular table:
%e 12, 18, 20, 24, 28, [36]; hence a(1) = 5,
%e 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, [72]; a(2) = 11,
%e 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, [100]; a(3) = 10,
%e 104, [108]; a(4) = 1,
%e 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, [144]; a(5) = 10, etc.
%t s = Select[Range[2^16], Nor[SquareFreeQ[#], PrimePowerQ[#]] &]; -1 + Length /@ TakeList[s, Prepend[Differences[#], First[#]] &@ Position[s, _Integer?(Divisible[#, Apply[Times, FactorInteger[#][[All, 1]] ]^2] &)][[All, 1]] ]
%Y Cf. A001694, A126706, A286708, A332785.
%K nonn,easy,new
%O 1,1
%A _Michael De Vlieger_, Dec 04 2024