A078514 Numbers k such that the smallest prime dividing k is the largest exponent in the factorization of k.

4, 12, 18, 20, 27, 28, 36, 44, 50, 52, 60, 68, 76, 84, 90, 92, 98, 100, 116, 124, 126, 132, 135, 140, 148, 150, 156, 164, 172, 180, 188, 189, 196, 198, 204, 212, 220, 228, 234, 236, 242, 244, 252, 260, 268, 276, 284, 292, 294, 297, 300, 306, 308, 316, 332, 338

Offset

1,1

Comments

Let d(k, m) be the asymptotic density of k-free numbers (numbers not divisible by a k-th power of a prime) that are not divisible by the first m primes. Then, d(k, m) = (1/zeta(k)) * Product_{j=1..m} (1-1/prime(j))/(1-1/prime(j)^k). The asymptotic density of terms whose smallest prime divisor is prime(i) is delta(i) = d(prime(i)+1, i-1) - d(prime(i), i-1) - d(prime(i)+1, i) + d(prime(i), i) and the asymptotic density of this sequence is Sum_{i>=1} delta(i) = 0.16785889468250175464... . - Amiram Eldar, Jun 24 2022

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Example

126 = 2 * 3^2 * 7 minimum prime is 2, largest exponent is also 2, hence 126 is in the sequence.

Mathematica

spleQ[n_]:=Module[{f=FactorInteger[n]}, f[[1, 1]]==Max[f[[All, 2]]]]; Select[ Range[2, 400], spleQ] (* Harvey P. Dale, Aug 19 2017 *)

Prog

(Python)
from sympy import factorint
def aupto(limit):
  alst = []
  for k in range(4, limit+1):
    f = factorint(k)
    if min(f) == max(f[p] for p in f): alst.append(k)
  return alst
print(aupto(338)) # Michael S. Branicky, Apr 12 2021

Crossrefs

Subsequence of A111371.

Sequence in context: A350602 A362873 A111371 * A324519 A230628 A074285

Adjacent sequences: A078511 A078512 A078513 * A078515 A078516 A078517

Keyword

nonn

Author

Benoit Cloitre, Jan 06 2003

Status

approved