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A378897 Number of integers that are neither squarefree nor prime powers between consecutive powerful numbers, exclusive of powerful numbers themselves.

%I A378897 #12 Dec 12 2024 15:24:44
%S A378897 0,0,0,1,3,0,1,0,4,6,1,3,7,1,4,1,1,4,10,9,1,4,2,6,5,11,0,12,8,7,12,1,
%T A378897 11,2,14,6,3,7,18,18,8,9,0,20,21,3,16,10,13,23,2,0,10,7,28,11,10,0,26,
%U A378897 26,8,3,7,5,0,26,30,17,11,32,20,13,12,20,36,1,20
%N A378897 Number of integers that are neither squarefree nor prime powers between consecutive powerful numbers, exclusive of powerful numbers themselves.
%C A378897 Also the number of terms in A126706 between powerful terms, exclusive of powerful terms. Therefore 36, which is both in A126706 and in A001694, is not counted.
%H A378897 Michael De Vlieger, <a href="/A378897/b378897.txt">Table of n, a(n) for n = 1..10000</a>
%F A378897 a(n) = A076446(n) - A378593(n) - 1 for n > 1.
%e A378897 Let s = A001694, powerful numbers.
%e A378897 Let t = A013929, nonsquarefree numbers.
%e A378897 a(1..3) = 0 since t(1) = 12 while s(1) = 1, s(2) = 4, s(3) = 8, and s(4) = 9.
%e A378897 a(4) = 1 since s(5) < t(1) < s(6), i.e., 9 < 12 < 16.
%e A378897 a(5) = 3 since between s(6) = 16 and s(7) = 25, we have t(2..4) = {18, 20, 24}.
%e A378897 a(6) = 0 since s(7) < 26 < s(8), where s(8) = 27, and 26 is squarefree.
%e A378897 a(7) = 1 since s(8) < t(5) < s(9), where t(5) = 28 and s(9) = 32,
%e A378897 a(8) = 0 since there are no nonsquarefree numbers between s(9) = 32 and s(10) = 36, etc.
%t A378897 With[{nn = 2^12},  s = Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}];  Table[Count[Range[s[[i]] + 1, s[[i + 1]] - 1],     _?(Not@*SquareFreeQ)], {i, Length[s] - 1}] ]
%Y A378897 Cf. A001694, A013929, A076446, A126706, A378593.
%K A378897 nonn
%O A378897 1,5
%A A378897 _Michael De Vlieger_, Dec 10 2024