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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.

%I A336223 #11 Jul 14 2020 03:37:19
%S A336223 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,
%T A336223 38,39,41,42,43,46,47,51,53,55,57,58,59,61,62,65,66,67,69,70,71,73,74,
%U A336223 77,78,79,82,83,85,86,87,89,91,93,94,95,97,100,101,102,103
%N 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.
%C A336223 First differs from A333634 at n = 47.
%C A336223 Terms k of A335275 such that A000188(k) is a term of A030231.
%C A336223 Numbers whose powerful part (A057521) is a square term of A030231.
%C A336223 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.
%C A336223 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).
%H A336223 Amiram Eldar, <a href="/A336223/b336223.txt">Table of n, a(n) for n = 1..10000</a>
%H A336223 Eckford Cohen, <a href="https://doi.org/10.1090/S0002-9947-1964-0166181-5">Some asymptotic formulas in the theory of numbers</a>, Trans. Amer. Math. Soc., Vol. 112 (1964), pp. 214-227.
%e A336223 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.
%t A336223 seqQ[n_] := EvenQ @ Length[(e = Select[FactorInteger[n][[;; , 2]], # > 1 &])] && AllTrue[e, EvenQ[#] &]; Select[Range[100], seqQ]
%Y A336223 Intersection of A333634 and A335275.
%Y A336223 Cf. A000188, A001221, A005117, A030231, A057521, A065465.
%K A336223 nonn
%O A336223 1,2
%A A336223 _Amiram Eldar_, Jul 12 2020