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

Author

Amiram Eldar, May 23 2020

Keywords

• nonn

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).

Examples

			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.
		

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.

Crossrefs

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

Programs

Mathematica

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

Formula

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