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

Author

Reinhard Zumkeller, Aug 21 2006

Keywords

• nonn
• easy

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.

Crossrefs

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

Complement: A038838.

Cf. A217739.

Programs

Haskell

a122132 n = a122132_list !! (n-1)
a122132_list = filter ((== 1) . a008966 . a000265) [1..]
-- Reinhard Zumkeller, Jan 24 2012

Mathematica

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

PARI

is(n)=issquarefree(n>>valuation(n,2)); \\ Charles R Greathouse IV, Sep 02 2015

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