A336224 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root 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, 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, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100

Offset: 1

Author

Amiram Eldar, Jul 12 2020

Keywords

• nonn
• easy

Comments

Terms k of A335275 such that A000188(k) is a term of A028260.

Numbers whose powerful part (A057521) is the square of a term of A028260.

The squarefree numbers (A005117) are terms of this sequence since if k is squarefree, then the largest square dividing k is 1 which is a unitary divisor, sqrt(1) has 0 prime divisors, and 0 is even.

The asymptotic density of this sequence is (5 * Product_{p prime} (1 - 1/(p^2*(p+1))) + 2 * Product_{p prime} (1 + 1/(p^2*(p+1))))/10 = (5 * 0.881513... + 2 * 1.125606...)/10 = 0.665878294481337275662425136416469977597382409701642... (Cohen, 1964; the first product is A065465).

Examples

			16 is a term since the largest square dividing 16 is 16, which is a unitary divisor, 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

Intersection of A335275 and A336222.

Cf. A000188, A001222, A005117, A028260, A057521, A065465.

Programs

Mathematica

seqQ[n_] := AllTrue[(e = FactorInteger[n][[;; , 2]]), # == 1 || EvenQ[#] &] && EvenQ @ Total[Select[e, # > 1 &]/2]; Select[Range[100], seqQ]