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

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, 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, 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

Offset: 1

Author

Reinhard Zumkeller, Nov 07 2004

Keywords

• nonn
• easy

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Squarefree

Crossrefs

Cf. A005117, A008966, A013928.

Programs

Mathematica

a[n_] := Sum[MoebiusMu[LCM[n, k]]^2, {k, 1, n}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 18 2021, after Benoit Cloitre *)

PARI

a(n)=sum(k=1,n,moebius(lcm(n,k))^2) \\ Benoit Cloitre, Jun 14 2007

PARI

first(n)=my(v=vector(n),k); forsquarefree(m=1,n, v[m[1]]=k++); v \\ Charles R Greathouse IV, Jan 08 2018

Python

from sympy import integer_nthroot, mobius
def A100112(n):
    b = 0
    k, r = integer_nthroot(n,2)
    a, c = (mobius(k),k-1) if r else (0,k)
    for i in range(1,c+1):
        m, j = mobius(i), i**2
        a += m*(n//j)
        b += m*((n-1)//j)
    return a if a>b else 0 # Chai Wah Wu, May 11 2024

Formula

a(A005117(n)) = n;

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

a(n) = Sum_{k=1..n} mu(lcm(n,k))^2. - Benoit Cloitre, Jun 14 2007