A388044 a(n) = number of k <= m such that rad(k) | m, where m = A001694(n) and rad(n) = A007947(n).

1, 3, 4, 3, 5, 3, 4, 6, 14, 3, 7, 18, 5, 15, 21, 3, 4, 8, 23, 3, 16, 19, 26, 13, 6, 9, 29, 3, 30, 4, 3, 20, 23, 32, 14, 18, 24, 10, 3, 35, 5, 36, 18, 19, 7, 24, 28, 3, 39, 83, 3, 21, 40, 29, 11, 15, 20, 42, 21, 13, 43, 18, 4, 22, 3, 27, 21, 15, 28, 33, 3, 46, 91

Offset

1,2

Comments

Cardinality of row A001694(n) of A162306.

Links

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

Michael De Vlieger, Efficient Mathematica program to generate large datasets.

Formula

a(n) = A010846(A001694(n)).

Let s = A001694 and let f = A010846.

For s(k) = p^m, p prime, f(s(k)) = m+1. Therefore, for n > 1, a(n) >= 3.

Example

Let R(k) = {j : rad(j) | k}. For example, R(6) = A003586.
a(1) = 1 since 1 is the smallest number such that rad(1) | 1.
a(2) = 3 since s(2) = 4 = 2^2, with f(2^2) = 3.
a(3) = 4 since s(3) = 8 = 2^3, with f(2^3) = 4.
a(4) = 3 since s(4) = 9 = 3^2, with f(3^2) = 3.
a(5) = 5 since s(5) = 16 = 2^4, with f(2^4) = 5.
a(9) = 14 since s(9) = 36, the 14th number in R(rad(36)) = R(6) = A003586.
a(12) = 18 since s(12) = 72, the 18th number in R(rad(72)) = R(6) = A003586.
a(14) = 15 since s(14) = 100, the 14th number in R(rad(100)) = R(10) = A003592, etc.

Mathematica

With[{nn = 2000}, Map[Function[k, Count[Range[k], _?(Divisible[k, rad[#]] &)]], Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]} ] ] ]

Crossrefs

Cf. A001694, A010846, A162306.

Sequence in context: A090739 A076400 A363194 * A121889 A332026 A205692

Adjacent sequences: A388041 A388042 A388043 * A388045 A388046 A388047

Keyword

nonn

Author

Michael De Vlieger, Sep 13 2025

Status

approved