A388306 Numbers that are not square but divisible by the square of their largest prime factor.

8, 18, 27, 32, 50, 54, 72, 75, 98, 108, 125, 128, 147, 150, 162, 200, 216, 242, 243, 245, 250, 288, 294, 300, 338, 343, 363, 375, 392, 432, 450, 486, 490, 500, 507, 512, 578, 588, 600, 605, 648, 675, 686, 722, 726, 735, 750, 800, 845, 847, 864, 867, 882, 968, 972, 980

Offset

1,1

Comments

From Michael De Vlieger, Sep 21 2025: (Start)

Let g = gpf(k) = A006530(k) be the greatest prime factor of k. Then a(n) = k is such that g^2 | k, and terms in this sequence cannot be squarefree, thus this sequence is a subset of A013929.

Furthermore, k >= rad(k)*g, where rad = A007947, and A003557(k) = k/rad(k) exceeds the second smallest prime factor A119288(k) of k. As a consequence regarding terms in A024619, this sequence does not intersect A366825 (i.e., m = lpf(k)*k with k in A120944, implies m in A126706, where lpf = A020639). This sequence also does not intersect A360967 (i.e., numbers k in A126706 such that A003557(k) < A119288(k)).

Since terms k are not square and since gpf(k)^2 | k, prime powers p^m in this sequence are such that m > 1 is odd. Therefore p^m is cubefull (i.e., in A246549), and this sequence is a superset of A056824. (End)

Links

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

Formula

k is term <=> k has canonical prime factorization p1^e1 *...* pj^ej with ej >= 2 and some ei in e1...ej is odd.

Mathematica

Select[Range[2^10], And[! IntegerQ[Sqrt[#]], Divisible[#, Power[FactorInteger[#][[-1, 1]], 2]]] &] (* Michael De Vlieger, Sep 21 2025 *)

Prog

(SageMath)
def isA388306(n: int) -> int: return not is_square(n) and factor(n)[-1][1] > 1
print([n for n in range(1000) if isA388306(n)])
(PARI) isok(k) = !issquare(k) && !(k % vecmax(factor(k)[, 1])^2); \\ Michel Marcus, Sep 21 2025

Crossrefs

Subsequence of A070003 and of A388654.

Cf. A000037, A006530, A013929, A056824 (prime powers in this sequence), A126706, A246549.

Sequence in context: A298161 A195419 A110188 * A069236 A187101 A051370

Adjacent sequences: A388303 A388304 A388305 * A388307 A388308 A388309

Keyword

nonn

Author

Peter Luschny, Sep 21 2025

Status

approved