A360158 a(n) is the number of unitary divisors of n that are odd squares minus the number of unitary divisors of n that are even squares.
1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
f[p_, e_] := If[OddQ[e], 1, 2]; f[2, e_] := If[OddQ[e], 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
-
PARI
a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, 1, if(f[i, 1] == 2, 0, 2)));}
Formula
a(n) = Sum_{d|n, gcd(d, n/d)=1, d square} (-1)^(d+1).
Multiplicative with a(2^e) = 1 if e is odd and 0 if e is even, and for p > 2, a(p^e) = 1 if e is odd and 2 if e is even.
Dirichlet g.f.: (zeta(s)*zeta(2*s)/zeta(3*s)) * (4^s + 2^s - 1)/(4^s + 2^s + 1).
Sum_{k=1..n} a(k) ~ c * n, where c = 5*zeta(2)/(7*zeta(3)) = 0.977451984014... .
Comments