A336223 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root has an even number of distinct 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, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100, 101, 102, 103

Offset

1,2

Comments

First differs from A333634 at n = 47.

Terms k of A335275 such that A000188(k) is a term of A030231.

Numbers whose powerful part (A057521) is a square term of A030231.

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) = 1 has 0 prime divisors, and 0 is even.

The asymptotic density of this sequence is (Product_{p prime} (1 - 1/(p^2*(p+1))) + Product_{p prime} (1 - (2*p+1)/(p^2*(p+1))))/2 = (0.881513... + 0.394391...)/2 = 0.637952807730728551636349961980617856650450613867264... (Cohen, 1964; the first product is A065465).

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

36 is a term since the largest square dividing 36 is 36, which is a unitary divisor, sqrt(36) = 6, 6 = 2 * 3 has 2 distinct prime divisors, and 2 is even.

Mathematica

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

Crossrefs

Intersection of A333634 and A335275.

Cf. A000188, A001221, A005117, A030231, A057521, A065465.

Sequence in context: A193304 A333634 A348499 * A348961 A367801 A348506

Adjacent sequences: A336220 A336221 A336222 * A336224 A336225 A336226

Keyword

nonn

Author

Amiram Eldar, Jul 12 2020

Status

approved