cp's OEIS Frontend

A336222 Numbers k such that the square root of the largest square dividing k has an even number of prime divisors (counted with multiplicity).

%I A336222 #11 Jul 14 2020 03:39:06
%S A336222 1,2,3,5,6,7,10,11,13,14,15,16,17,19,21,22,23,26,29,30,31,32,33,34,35,
%T A336222 36,37,38,39,41,42,43,46,47,48,51,53,55,57,58,59,61,62,65,66,67,69,70,
%U A336222 71,72,73,74,77,78,79,80,81,82,83,85,86,87,89,91,93,94,95
%N A336222 Numbers k such that the square root of the largest square dividing k has an even number of prime divisors (counted with multiplicity).
%C A336222 Numbers k such that A000188(k) is a term of A028260.
%C A336222 The squarefree numbers (A005117) are terms of this sequence since if k is squarefree, then A000188(k) = 1, 1 has 0 prime divisors, and 0 is even.
%C A336222 A number k is a term if and only if its powerful part, A057521(k), is a term.
%C A336222 The asymptotic density of this sequence is 7/10 (Cohen, 1964).
%C A336222 The corresponding sequence of numbers k such that the square root of the largest square dividing k has an even number of distinct prime divisors, i.e., numbers k such that A000188(k) is a term of A030231, is A333634.
%H A336222 Amiram Eldar, <a href="/A336222/b336222.txt">Table of n, a(n) for n = 1..10000</a>
%H A336222 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 A336222 2 is a term since the largest square dividing 2 is 1, sqrt(1) = 1, 1 has 0 prime divisors, and 0 is even.
%e A336222 16 is a term since the largest square dividing 16 is 16, sqrt(16) = 4, 4 = 2 * 2 has 2 prime divisors, and 2 is even.
%t A336222 f[p_, e_] := p^Floor[e/2]; Select[Range[100], EvenQ[PrimeOmega[Times @@ (f @@@ FactorInteger[#])]] &]
%Y A336222 Cf. A000188, A001222, A005117, A028260, A030231, A057521, A333634.
%K A336222 nonn
%O A336222 1,2
%A A336222 _Amiram Eldar_, Jul 12 2020