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

1,5

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.

Links

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

Formula

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

Example

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.

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}] ]

Crossrefs

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

Sequence in context: A329861 A331332 A300228 * A100573 A049087 A178921

Adjacent sequences: A378894 A378895 A378896 * A378898 A378899 A378900

Keyword

nonn

Author

Michael De Vlieger, Dec 10 2024

Status

approved