A333634 Numbers with an even number of non-unitary prime divisors.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100, 101, 102

Offset

1,2

Comments

Numbers that have an even number of distinct prime factors raised to a power larger than 1.

The asymptotic density of this sequence is 0.661317... (A065493, Feller and Tornier, 1933).

Links

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

Willy Feller and Erhard Tornier, Mengentheoretische Untersuchung von Eigenschaften der Zahlenreihe, Mathematische Annalen, Vol. 107 (1933), pp. 188-232.

I. J. Schoenberg, On asymptotic distributions of arithmetical functions, Transactions of the American Mathematical Society, Vol. 39, No. 2 (1936), pp. 315-330. See p. 326.

Eric Weisstein's World of Mathematics, Feller-Tornier Constant.

Wikipedia, Feller-Tornier constant.

Formula

Numbers k with A056170(k) == 0 (mod 2).

Example

1 is a term since it has 0 prime divisors, and 0 is even.
180 is a term since 180 = 2^2 * 3^2 * 5 has 2 prime divisors, 2 and 3, with exponents larger than 1 in its prime factorization, and 2 is even.

Mathematica

Select[Range[100], EvenQ @ Count[FactorInteger[#][[;; , 2]], u_ /; u > 1]  &]

Crossrefs

Cf. A056170, A065493, A190641, A327877 (complement).

Sequence in context: A325511 A240370 A193304 * A348499 A336223 A348961

Adjacent sequences: A333631 A333632 A333633 * A333635 A333636 A333637

Keyword

nonn

Author

Amiram Eldar, May 23 2020

Status

approved