A111371 A number k is included if at least one prime dividing k is equal to an exponent of the highest power of any prime dividing k.

4, 12, 18, 20, 24, 27, 28, 36, 44, 50, 52, 54, 60, 68, 72, 76, 84, 90, 92, 98, 100, 108, 116, 120, 124, 126, 132, 135, 140, 144, 148, 150, 156, 160, 164, 168, 172, 180, 188, 189, 196, 198, 200, 204, 212, 216, 220, 228, 234, 236, 242, 244, 252, 260, 264, 268, 270

Offset

1,1

Comments

The number of terms not exceeding 10^m, for m = 1, 2, ..., are 1, 21, 216, 2186, 21921, 219274, 2192979, 21930103, 219301557, 2193017386, ... . Apparently, the asymptotic density of this sequence exists and equals 0.21930... . - Amiram Eldar, Jun 24 2022

Links

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

Example

50 = 2^1 * 5^2. 2 is both a prime dividing 50 and the exponent of the highest power of 5 dividing 50. So 50 is in the sequence.
144 = 2^4 * 3^2.  2 is a prime dividing 144 and the exponent of the highest power of 3 dividing 144.  So 144 is in the sequence.

Mathematica

Select[Range[2, 300], Intersection @@ Transpose[FactorInteger[ # ]] != {} &] (* Ray Chandler, Nov 09 2005 *)

Prog

(Python)
from sympy import factorint
def aupto(limit):
  alst = []
  for k in range(4, limit+1):
    f = factorint(k)
    # if max(f[p] for p in f) in f: alst.append(k)
    if set(f[p] for p in f) & set(f) != set(): alst.append(k)
  return alst
print(aupto(270)) # Michael S. Branicky, Apr 12 2021

Crossrefs

Supersequence of A078514.

Sequence in context: A175704 A350602 A362873 * A078514 A324519 A230628

Adjacent sequences: A111368 A111369 A111370 * A111372 A111373 A111374

Keyword

nonn

Author

Leroy Quet, Nov 08 2005

Extensions

Extended by Ray Chandler and Robert G. Wilson v, Nov 09 2005

Wrong Mathematica code removed by Amiram Eldar, Jun 24 2022

Status

approved