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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.
2
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
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.
Sequence in context: A193304 A333634 A348499 * A348961 A367801 A348506
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jul 12 2020
STATUS
approved