A122132 Squarefree numbers multiplied by binary powers.

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 85

Offset

1,2

Comments

These numbers are called "oddly squarefree" in Banks and Luca. - Michel Marcus, Mar 14 2016

The asymptotic density of this sequence is 8/Pi^2 (A217739). - Amiram Eldar, Sep 21 2020

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

William D. Banks and Florian Luca, Roughly squarefree values of the Euler and Carmichael functions, Acta Arithmetica, Vol. 120, No. 3 (2005), pp. 211-230.

Formula

a(n) = A007947(a(n)) * A006519(a(n)) / (2 - a(n) mod 2);

A007947(a(n)) = A000265(a(n)) * (2 - a(n) mod 2).

A008966(A000265(a(n))) = 1. - Reinhard Zumkeller, Jan 24 2012

A010052(A008477(a(n))) = 1. - Reinhard Zumkeller, Feb 17 2012

Mathematica

Select[Range@ 85, SquareFreeQ[#/2^IntegerExponent[#, 2]] &] (* Michael De Vlieger, Mar 15 2020 *)

Prog

(Haskell)
a122132 n = a122132_list !! (n-1)
a122132_list = filter ((== 1) . a008966 . a000265) [1..]
-- Reinhard Zumkeller, Jan 24 2012
(PARI) is(n)=issquarefree(n>>valuation(n, 2)); \\ Charles R Greathouse IV, Sep 02 2015

Crossrefs

Subsequences: A000079, A005117, A007283, A051916, A056911, A029747.

Complement: A038838.

Cf. A217739.

Sequence in context: A043094 A023803 A353511 * A347248 A347243 A325389

Adjacent sequences: A122129 A122130 A122131 * A122133 A122134 A122135

Keyword

nonn,easy

Author

Reinhard Zumkeller, Aug 21 2006

Status

approved