cp's OEIS Frontend

A346014 Numbers whose average number of distinct prime factors of their divisors is an integer.

%I A346014 #8 Jul 02 2021 03:49:36
%S A346014 1,6,10,14,15,21,22,26,33,34,35,38,39,46,51,55,57,58,62,65,69,74,77,
%T A346014 82,85,86,87,91,93,94,95,106,111,115,118,119,122,123,129,133,134,141,
%U A346014 142,143,145,146,155,158,159,161,166,177,178,183,185,187,194,201,202
%N A346014 Numbers whose average number of distinct prime factors of their divisors is an integer.
%C A346014 First differs from A030229 at n = 275. a(275) = 900 is the least term that is not squarefree and therefore not in A030229.
%C A346014 The least term whose exponents in its prime factorization are not all the same is 1080 = 2^3 * 3^3 * 5.
%C A346014 The least term whose exponents in its prime factorization are distinct is 1440 = 2^5 * 3^2 * 5.
%C A346014 Numbers k such that A000005(k) | A062799(k).
%C A346014 Numbers k such that A346010(k) = 1.
%C A346014 Numbers k such that if the prime factorization of k is Product_{i} p_i^e_i, then Sum_{i} e_i/(e_i + 1) is an integer.
%C A346014 Includes all the squarefree numbers with an even number of prime divisors (A030229), i.e., the union of A006881, A046386, A067885, A123322, ...
%C A346014 If k is squarefree with m prime divisors then k^(m-1) is a term. E.g., the squares of the sphenic numbers (A162143) are terms.
%H A346014 Amiram Eldar, <a href="/A346014/b346014.txt">Table of n, a(n) for n = 1..10000</a>
%e A346014 6 is a term since it has 4 divisors, 1, 2, 3 and 6 and (omega(1) + omega(2) + omega(3) + omega(6))/4 = (0 + 1 + 1 + 2)/4 = 1 is an integer.
%t A346014 f[p_, e_] := e/(e + 1); d[1] = 1; d[n_] := Denominator[Plus @@ f @@@ FactorInteger[n]]; Select[Range[200], d[#] == 1 &]
%Y A346014 Cf. A000005, A001221, A062799, A346009, A346010.
%Y A346014 Subsequences: A006881, A030229, A046386, A067885, A123322, A162143.
%K A346014 nonn
%O A346014 1,2
%A A346014 _Amiram Eldar_, Jul 01 2021