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%I #82 Dec 03 2022 05:47:16
%S 8,12,18,20,24,27,28,32,40,44,45,48,50,52,54,56,60,63,68,72,75,76,80,
%T 84,88,90,92,96,98,99,104,108,112,116,117,120,124,125,126,128,132,135,
%U 136,140,147,148,150,152,153,156,160,162,164,168,171,172,175,176,180
%N Nonsquarefree nonsquares: each term has a square factor but is not a perfect square itself.
%C 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
%C 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
%H Reinhard Zumkeller, <a href="/A051144/b051144.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareNumber.html">Square Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Square_number">Square number</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Squarefree_integer">Squarefree integer</a>.
%F (1 - A008966(a(n)))*(1 - A010052(a(n))) = 1; A008966(a(n)) + A010052(a(n)) = 0. - _Reinhard Zumkeller_, Jan 24 2013
%F 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
%e 63 is included because 63 = 3^2 * 7.
%e 64 is not included because it is a perfect square (8^2).
%e 65 is not included because it is squarefree (5 * 13).
%p N:= 10000; # to get all entries up to N
%p A051144:= remove(numtheory:-issqrfree,{$1..N}) minus {seq(i^2,i=1..floor(sqrt(N)))}:
%p # _Robert Israel_, Mar 30 2014
%t searchMax = 32; Complement[Select[Range[searchMax^2], MoebiusMu[#] == 0 &], Range[searchMax]^2] (* _Alonso del Arte_, Dec 20 2019 *)
%o (Haskell)
%o a051144 n = a051144_list !! (n-1)
%o a051144_list = filter ((== 0) . a008966) a000037_list
%o -- _Reinhard Zumkeller_, Sep 02 2013, Jan 24 2013
%o (PARI) is(n)=!issquare(n) && !issquarefree(n) \\ _Charles R Greathouse IV_, Sep 18 2015
%o (Magma) [k:k in [1..200]| not IsSquare(k) and not IsSquarefree(k)]; // _Marius A. Burtea_, Dec 29 2019
%Y Cf. A210490 (complement), intersection of A013929 and A000037.
%Y Related to A177712 via A225546.
%K nonn,easy
%O 1,1
%A Michael Minic (Rassilon6(AT)aol.com)
%E Incorrect comment removed by _Charles R Greathouse IV_, Mar 19 2010
%E Offset corrected by _Reinhard Zumkeller_, Jan 24 2013
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