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A078514 Numbers k such that the smallest prime dividing k is the largest exponent in the factorization of k. 2
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Jan 06 2003
STATUS
approved

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Last modified February 25 18:54 EST 2024. Contains 370332 sequences. (Running on oeis4.)