A365404 The sum of the unitary divisors of the square root of the largest square dividing n.
1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 5, 1, 4, 1, 3, 1, 1, 1, 3, 6, 1, 4, 3, 1, 1, 1, 5, 1, 1, 1, 12, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 5, 8, 6, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 9, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 6, 3, 1, 1, 1, 5, 10, 1, 1, 3, 1, 1
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
-
Mathematica
f[p_, e_] := If[e == 1, 1, p^Floor[e/2] + 1]; 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] == 1, 1, 1 + f[i,1]^(f[i,2]\2)));}
Formula
a(n) >= 1 with equality if and only if n is squarefree (A005117).
Multiplicative with a(p) = 1 and a(p^e) = p^floor(e/2) + 1 for e >= 2.
Dirichlet g.f.: zeta(s) * zeta(2*s-1) / zeta(4*s-1).
Sum_{k=1..n} a(k) ~ (n/(2*zeta(3))) * (log(n) + 3*gamma - 1 - 4*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620).
Comments