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A100112 If n is the k-th squarefree number then a(n) = k, otherwise a(n) = 0.

%I A100112 #27 Feb 16 2025 08:32:55
%S A100112 1,2,3,0,4,5,6,0,0,7,8,0,9,10,11,0,12,0,13,0,14,15,16,0,0,17,0,0,18,
%T A100112 19,20,0,21,22,23,0,24,25,26,0,27,28,29,0,0,30,31,0,0,0,32,0,33,0,34,
%U A100112 0,35,36,37,0,38,39,0,0,40,41,42,0,43,44,45,0,46,47,0,0,48,49,50,0,0,51,52,0
%N A100112 If n is the k-th squarefree number then a(n) = k, otherwise a(n) = 0.
%H A100112 Charles R Greathouse IV, <a href="/A100112/b100112.txt">Table of n, a(n) for n = 1..10000</a>
%H A100112 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%F A100112 a(A005117(n)) = n;
%F A100112 a(n) = (A013928(n) + 1)*A008966(n).
%F A100112 a(n) = Sum_{k=1..n} mu(lcm(n,k))^2. - _Benoit Cloitre_, Jun 14 2007
%t A100112 a[n_] := Sum[MoebiusMu[LCM[n, k]]^2, {k, 1, n}];
%t A100112 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Nov 18 2021, after _Benoit Cloitre_ *)
%o A100112 (PARI) a(n)=sum(k=1,n,moebius(lcm(n,k))^2) \\ _Benoit Cloitre_, Jun 14 2007
%o A100112 (PARI) first(n)=my(v=vector(n),k); forsquarefree(m=1,n, v[m[1]]=k++); v \\ _Charles R Greathouse IV_, Jan 08 2018
%o A100112 (Python)
%o A100112 from sympy import integer_nthroot, mobius
%o A100112 def A100112(n):
%o A100112     b = 0
%o A100112     k, r = integer_nthroot(n,2)
%o A100112     a, c = (mobius(k),k-1) if r else (0,k)
%o A100112     for i in range(1,c+1):
%o A100112         m, j = mobius(i), i**2
%o A100112         a += m*(n//j)
%o A100112         b += m*((n-1)//j)
%o A100112     return a if a>b else 0 # _Chai Wah Wu_, May 11 2024
%Y A100112 Cf. A005117, A008966, A013928.
%K A100112 nonn,easy
%O A100112 1,2
%A A100112 _Reinhard Zumkeller_, Nov 07 2004