A336222 Numbers k such that the square root of the largest square dividing k has an even number of prime divisors (counted with multiplicity).

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95

Offset

1,2

Comments

Numbers k such that A000188(k) is a term of A028260.

The squarefree numbers (A005117) are terms of this sequence since if k is squarefree, then A000188(k) = 1, 1 has 0 prime divisors, and 0 is even.

A number k is a term if and only if its powerful part, A057521(k), is a term.

The asymptotic density of this sequence is 7/10 (Cohen, 1964).

The corresponding sequence of numbers k such that the square root of the largest square dividing k has an even number of distinct prime divisors, i.e., numbers k such that A000188(k) is a term of A030231, is A333634.

Links

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

Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112 (1964), pp. 214-227.

Example

2 is a term since the largest square dividing 2 is 1, sqrt(1) = 1, 1 has 0 prime divisors, and 0 is even.
16 is a term since the largest square dividing 16 is 16, sqrt(16) = 4, 4 = 2 * 2 has 2 prime divisors, and 2 is even.

Mathematica

f[p_, e_] := p^Floor[e/2]; Select[Range[100], EvenQ[PrimeOmega[Times @@ (f @@@ FactorInteger[#])]] &]

Crossrefs

Cf. A000188, A001222, A005117, A028260, A030231, A057521, A333634.

Sequence in context: A101882 A318239 A059266 * A252895 A366242 A336224

Adjacent sequences: A336219 A336220 A336221 * A336223 A336224 A336225

Keyword

nonn

Author

Amiram Eldar, Jul 12 2020

Status

approved