|
|
A051144
|
|
Nonsquarefree nonsquares: each term has a square factor but is not a perfect square itself.
|
|
6
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
Eric Weisstein's World of Mathematics, Squarefree.
|
|
FORMULA
|
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)))}:
|
|
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
(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
|
|
|
STATUS
|
approved
|
|
|
|