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

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202

Offset: 1

Author

Amiram Eldar, Jul 01 2021

Keywords

• nonn

Comments

First differs from A030229 at n = 275. a(275) = 900 is the least term that is not squarefree and therefore not in A030229.

The least term whose exponents in its prime factorization are not all the same is 1080 = 2^3 * 3^3 * 5.

The least term whose exponents in its prime factorization are distinct is 1440 = 2^5 * 3^2 * 5.

Numbers k such that A000005(k) | A062799(k).

Numbers k such that A346010(k) = 1.

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.

Includes all the squarefree numbers with an even number of prime divisors (A030229), i.e., the union of A006881, A046386, A067885, A123322, ...

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A000005, A001221, A062799, A346009, A346010.

Subsequences: A006881, A030229, A046386, A067885, A123322, A162143.

Programs

Mathematica

f[p_, e_] := e/(e + 1); d[1] = 1; d[n_] := Denominator[Plus @@ f @@@ FactorInteger[n]]; Select[Range[200], d[#] == 1 &]