A100112 - OEIS

%I #26 May 11 2024 02:24:55

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

%H Charles R Greathouse IV, <a href="/A100112/b100112.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>

%F a(A005117(n)) = n;

%F a(n) = (A013928(n) + 1)*A008966(n).

%F a(n) = Sum_{k=1..n} mu(lcm(n,k))^2. - _Benoit Cloitre_, Jun 14 2007

%t a[n_] := Sum[MoebiusMu[LCM[n, k]]^2, {k, 1, n}];

%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Nov 18 2021, after _Benoit Cloitre_ *)

%o (PARI) a(n)=sum(k=1,n,moebius(lcm(n,k))^2) \\ _Benoit Cloitre_, Jun 14 2007

%o (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 (Python)

%o from sympy import integer_nthroot, mobius

%o def A100112(n):

%o     b = 0

%o     k, r = integer_nthroot(n,2)

%o     a, c = (mobius(k),k-1) if r else (0,k)

%o     for i in range(1,c+1):

%o         m, j = mobius(i), i**2

%o         a += m*(n//j)

%o         b += m*((n-1)//j)

%o     return a if a>b else 0 # _Chai Wah Wu_, May 11 2024

%Y Cf. A005117, A008966, A013928.

%K nonn,easy

%O 1,2

%A _Reinhard Zumkeller_, Nov 07 2004