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

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]
def isA383264(n: int) -> bool: return not is_squarefree(vSPD(n))
print([n for n in range(2, 1300) if isA383264(n)])
(PARI) isok(k) = if (k>1, !issquarefree(factor(k)[1, 2])); \\ Michel Marcus, Apr 25 2025

Crossrefs

Cf. A067029, A013929, A008683.

Sequence in context: A392279 A336595 A386804 * A069084 A084112 A050428

Adjacent sequences: A383261 A383262 A383263 * A383265 A383266 A383267

Keyword

nonn,easy

Author

Peter Luschny, Apr 25 2025

Status

approved