A378593 - OEIS

%I #13 Dec 09 2024 23:25:13

%S 3,3,0,5,5,1,3,3,8,8,6,5,11,6,8,2,1,11,14,17,2,11,6,11,7,20,0,22,10,

%T 10,18,6,20,6,28,9,8,9,28,30,14,17,0,32,33,12,24,12,22,37,4,3,17,16,

%U 36,24,16,3,42,44,18,4,13,11,2,45,46,29,20,48,26,22,23

%N Number of squarefree k between consecutive powerful numbers (inclusive).

%H Michael De Vlieger, <a href="/A378593/b378593.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) > A076446(n) for n > 1.

%F a(n) >= A240590(n).

%e a(1) = 3 since {[1], 2, 3, [4]} has 3 squarefree numbers.

%e a(2) = 3 since {[4], 5, 6, 7, [8]} has 3 squarefree numbers.

%e a(3) = 0 since {[8], [9]} has no squarefree numbers.

%e a(4) = 5 since between 9 and 16, {10, 11, 13, 14, 15} are squarefree.

%e a(5) = 5 since between 16 and 25, {17, 19, 21, 22, 23} are squarefree, etc.

%t With[{nn = 2^12},

%t   s = Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}];  Table[Count[Range[s[[i]], s[[i + 1]]], _?SquareFreeQ], {i, Length[s] - 1}]

%Y Cf. A001694, A005117, A076446, A240590.

%K nonn

%O 1,1

%A _Michael De Vlieger_, Dec 09 2024