Voozh

A383264

Numbers whose vSPD is not squarefree, where vSPD(n) is the valuation of the smallest prime divisor for n >= 2.

16, 48, 80, 81, 112, 144, 176, 208, 240, 256, 272, 304, 336, 368, 400, 405, 432, 464, 496, 512, 528, 560, 567, 592, 624, 625, 656, 688, 720, 752, 768, 784, 816, 848, 880, 891, 912, 944, 976, 1008, 1040, 1053, 1072, 1104, 1136, 1168, 1200, 1232, 1264, 1280, 1296

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The asymptotic density of this sequence is Sum_{k>=1} (Product_{i=1..k-1} (1 - 1/prime(i)) * Sum_{j>=4} (mu(j-1)^2 - mu(j)^2)/prime(k)^j) = 0.03904378342010..., where mu is the Möbius function (A008683). - Amiram Eldar, Apr 26 2025

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

MAPLE

with(NumberTheory):

espf := n -> ifelse(n = 1, 1, ifactors(n)[2][1][2]): # modified A067029

isA383264 := n -> is(Moebius(espf(n)) = 0):

select(isA383264, [seq(1..1300)]); # Peter Luschny, Jun 19 2025

MATHEMATICA

Select[Range[2, 1296], ! SquareFreeQ@ FactorInteger[#][[1, 2]] &] (* Michael De Vlieger, Apr 25 2025 *)

PROG

(SageMath)

def vSPD(n: int) -> int: return factor(n)[0][1]