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

1,2

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.

Links

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

Example

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.

Mathematica

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

Crossrefs

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

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

Sequence in context: A211484 A339561 A350486 * A006881 A030229 A334342

Adjacent sequences: A346011 A346012 A346013 * A346015 A346016 A346017

Keyword

nonn

Author

Amiram Eldar, Jul 01 2021

Status

approved