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

Author

Amiram Eldar, Jul 12 2020

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

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

Programs

Mathematica

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