A051144 Nonsquarefree nonsquares: each term has a square factor but is not a perfect square itself.

8, 12, 18, 20, 24, 27, 28, 32, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180

Offset

1,1

Comments

At least one exponent in the canonical prime factorization (cf. A124010) of n is odd, and at least one exponent is greater than 1. - Reinhard Zumkeller, Jan 24 2013

Compare this sequence, as a set, with A177712, numbers that have an odd factor, but are not odd. The self-inverse function defined by A225546, maps the members of either one of these sets 1:1 onto the other set. - Peter Munn, Jul 31 2020

Links

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

Eric Weisstein's World of Mathematics, Square Number.

Eric Weisstein's World of Mathematics, Squarefree.

Wikipedia, Square number.

Wikipedia, Squarefree integer.

Formula

(1 - A008966(a(n)))*(1 - A010052(a(n))) = 1; A008966(a(n)) + A010052(a(n)) = 0. - Reinhard Zumkeller, Jan 24 2013

Sum_{n>=1} 1/a(n)^s = 1 + zeta(s) - zeta(2*s) - zeta(s)/zeta(2*s), for s > 1. - Amiram Eldar, Dec 03 2022

Example

63 is included because 63 = 3^2 * 7.
64 is not included because it is a perfect square (8^2).
65 is not included because it is squarefree (5 * 13).

Maple

N:= 10000;  # to get all entries up to N
A051144:= remove(numtheory:-issqrfree, {$1..N}) minus {seq(i^2, i=1..floor(sqrt(N)))}:
# Robert Israel, Mar 30 2014

Mathematica

searchMax = 32; Complement[Select[Range[searchMax^2], MoebiusMu[#] == 0 &], Range[searchMax]^2] (* Alonso del Arte, Dec 20 2019 *)

Prog

(Haskell)
a051144 n = a051144_list !! (n-1)
a051144_list = filter ((== 0) . a008966) a000037_list
-- Reinhard Zumkeller, Sep 02 2013, Jan 24 2013
(PARI) is(n)=!issquare(n) && !issquarefree(n) \\ Charles R Greathouse IV, Sep 18 2015
(Magma) [k:k in [1..200]| not IsSquare(k) and not IsSquarefree(k)]; // Marius A. Burtea, Dec 29 2019

Crossrefs

Cf. A210490 (complement), intersection of A013929 and A000037.

Related to A177712 via A225546.

Sequence in context: A104201 A036327 A340681 * A067537 A046339 A145784

Adjacent sequences: A051141 A051142 A051143 * A051145 A051146 A051147

Keyword

nonn,easy

Author

Michael Minic (Rassilon6(AT)aol.com)

Extensions

Incorrect comment removed by Charles R Greathouse IV, Mar 19 2010

Offset corrected by Reinhard Zumkeller, Jan 24 2013

Status

approved