A390538 Numbers that do not exceed the square of their squarefree kernel.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76

Offset

1,2

Comments

Numbers k such that k <= rad(k)^2, where rad = A007947.

Complement of A059172.

Union of {1}, A177492, and A341646.

Define sequence S(r) to be the set {m*r : rad(m) | r, m >= 1} for squarefree r (i.e., r in A005117). Then S(r) = r * {m : rad(m) | r} and so we have all terms in S(r) that do not exceed r^2 in this sequence. This is to say, given S(r,j) is the j-th term in S(r), that this sequence contains S(r,j) for j = 1..A010846(r). Therefore this sequence is a superset of A120944.

For prime p, the intersection of this sequence and S(p) = {p^m : m >= 1} is {p, p^2}. Therefore, this sequence is a superset of primes (A000040) and prime squares (A001248).

Links

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

Example

For r = 1, S(1) = {1}, 1 does not exceed 1^2, therefore a(1) = 1.
For r = 2, this sequence contains S(r,j) = A000079(j), j = 1..A010846(2), i.e., {2, 4}.
For r = 6, this sequence contains S(r,j) = A033485(j), j = 1..A010846(6), i.e., {6, 12, 18, 24, 36}.
For r = 10, this sequence contains S(r,j) = A033486(j), j = 1..A010846(10), i.e., {10, 20, 40, 50, 80, 100}. Therefore, a(7) = 160 is the first term in this sequence that is not in A033845.

Mathematica

Select[Range[76], # <= Apply[Times, FactorInteger[#][[All, 1]]^2] &]

Crossrefs

Cf. A000040, A001248, A003557, A005117, A007947, A059172, A120944, A177492, A341646.

Sequence in context: A378287 A362147 A361395 * A329135 A182768 A368917

Adjacent sequences: A390535 A390536 A390537 * A390539 A390540 A390541

Keyword

nonn

Author

Michael De Vlieger, Nov 09 2025

Status

approved