A111371 - OEIS

%I #21 Jun 24 2022 05:29:04

%S 4,12,18,20,24,27,28,36,44,50,52,54,60,68,72,76,84,90,92,98,100,108,

%T 116,120,124,126,132,135,140,144,148,150,156,160,164,168,172,180,188,

%U 189,196,198,200,204,212,216,220,228,234,236,242,244,252,260,264,268,270

%N 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.

%C 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

%H Amiram Eldar, <a href="/A111371/b111371.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%e 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.

%t Select[Range[2, 300], Intersection @@ Transpose[FactorInteger[ # ]] != {} &] (* _Ray Chandler_, Nov 09 2005 *)

%o (Python)

%o from sympy import factorint

%o def aupto(limit):

%o   alst = []

%o   for k in range(4, limit+1):

%o     f = factorint(k)

%o     # if max(f[p] for p in f) in f: alst.append(k)

%o     if set(f[p] for p in f) & set(f) != set(): alst.append(k)

%o   return alst

%o print(aupto(270)) # _Michael S. Branicky_, Apr 12 2021

%Y Supersequence of A078514.

%K nonn

%O 1,1

%A _Leroy Quet_, Nov 08 2005

%E Extended by _Ray Chandler_ and _Robert G. Wilson v_, Nov 09 2005

%E Wrong Mathematica code removed by _Amiram Eldar_, Jun 24 2022