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
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