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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
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
1,2
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).
LINKS
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112 (1964), pp. 214-227.
EXAMPLE
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.
MATHEMATICA
seqQ[n_] := AllTrue[(e = FactorInteger[n][[;; , 2]]), # == 1 || EvenQ[#] &] && EvenQ @ Total[Select[e, # > 1 &]/2]; Select[Range[100], seqQ]
CROSSREFS
Intersection of A335275 and A336222.
Sequence in context: A336222 A252895 A366242 * A274034 A197680 A361177
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Jul 12 2020
STATUS
approved