A337052 Numbers k such that the powerful part of k has an even number of prime divisors counted with multiplicity.

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, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76

Offset

1,2

Comments

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

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

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

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

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...

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112, No. 2 (1964), pp. 214-227. See corollary 4.2.2, pp. 226-227.

Example

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

Mathematica

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

Crossrefs

Cf. A001222, A028260, A057521, A096432.

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

Sequence in context: A270420 A336592 A335275 * A220218 A096432 A369938

Adjacent sequences: A337049 A337050 A337051 * A337053 A337054 A337055

Keyword

nonn

Author

Amiram Eldar, Aug 12 2020

Status

approved