cp's OEIS Frontend

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).

%I A336224 #16 Jan 31 2024 20:28:01
%S A336224 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,
%T A336224 37,38,39,41,42,43,46,47,48,51,53,55,57,58,59,61,62,65,66,67,69,70,71,
%U A336224 73,74,77,78,79,80,81,82,83,85,86,87,89,91,93,94,95,97,100
%N 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).
%C A336224 Terms k of A335275 such that A000188(k) is a term of A028260.
%C A336224 Numbers whose powerful part (A057521) is the square of a term of A028260.
%C A336224 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.
%C A336224 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).
%H A336224 Amiram Eldar, <a href="/A336224/b336224.txt">Table of n, a(n) for n = 1..10000</a>
%H A336224 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 A336224 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.
%t A336224 seqQ[n_] := AllTrue[(e = FactorInteger[n][[;; , 2]]), # == 1 || EvenQ[#] &] && EvenQ @ Total[Select[e, # > 1 &]/2]; Select[Range[100], seqQ]
%Y A336224 Intersection of A335275 and A336222.
%Y A336224 Cf. A000188, A001222, A005117, A028260, A057521, A065465.
%K A336224 nonn,easy
%O A336224 1,2
%A A336224 _Amiram Eldar_, Jul 12 2020