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

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, 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, 26, 8, 3, 7, 5, 0, 26, 30, 17, 11, 32, 20, 13, 12, 20, 36, 1, 20

Offset: 1

Author

Michael De Vlieger, Dec 10 2024

Keywords

• nonn

Comments

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.

Examples

			Let s = A001694, powerful numbers.
Let t = A013929, nonsquarefree numbers.
a(1..3) = 0 since t(1) = 12 while s(1) = 1, s(2) = 4, s(3) = 8, and s(4) = 9.
a(4) = 1 since s(5) < t(1) < s(6), i.e., 9 < 12 < 16.
a(5) = 3 since between s(6) = 16 and s(7) = 25, we have t(2..4) = {18, 20, 24}.
a(6) = 0 since s(7) < 26 < s(8), where s(8) = 27, and 26 is squarefree.
a(7) = 1 since s(8) < t(5) < s(9), where t(5) = 28 and s(9) = 32,
a(8) = 0 since there are no nonsquarefree numbers between s(9) = 32 and s(10) = 36, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001694, A013929, A076446, A126706, A378593.

Programs

Mathematica

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}] ]

Formula

a(n) = A076446(n) - A378593(n) - 1 for n > 1.