login
Numbers k such that the powerful part of k has an even number of prime divisors counted with multiplicity.
3

%I #12 Aug 17 2020 02:05:45

%S 1,2,3,4,5,6,7,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,25,26,28,

%T 29,30,31,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,50,51,52,53,

%U 55,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,73,74,75,76

%N Numbers k such that the powerful part of k has an even number of prime divisors counted with multiplicity.

%C Numbers k such that A001222(A057521(k)) == 0 (mod 2).

%C Numbers k such that A057521(k) is in A028260.

%C Differs from A096432 by having the additional terms 1 and 216, 256, 768, 864, ... and not having the terms 432, 648, ...

%C First differs from both A220218 and A335275 at n = 193: a(193) = 216 is not a term of these two sequences.

%C Cohen (1964) proved that this sequence has an asymptotic density, and gave the value 1/2 + (1/5) * Product_{p prime} (1 + (p^2 + p + 1)/(p^3 * (p + 1))) = 0.8172707179... But the numbers of terms not exceeding 10^k for k = 1, 2, ... are 9, 90, 885, 8849, 88499, 884993, 8849889, 88498711, 884987643, 8849876178, ... indicating that the asymptotic density is about 0.88498...

%H Amiram Eldar, <a href="/A337052/b337052.txt">Table of n, a(n) for n = 1..10000</a>

%H 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, No. 2 (1964), pp. 214-227. See corollary 4.2.2, pp. 226-227.

%e 2 is a term since the powerful part of 2 is 1, which has 0 prime divisors, and 0 is even.

%t Select[Range[100], EvenQ @ Total @ Select[FactorInteger[#][[;; , 2]], #1 > 1 &] &]

%Y Cf. A001222, A028260, A057521, A096432.

%Y Subsequences: A000290, A004709, A005117, A138302, A220218, A335275 and A336615.

%K nonn

%O 1,2

%A _Amiram Eldar_, Aug 12 2020