A360162 a(n) is the sum of the square roots of the unitary divisors of n that are squares.

1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 1, 3, 1, 1, 1, 5, 1, 4, 1, 3, 1, 1, 1, 1, 6, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 5, 8, 6, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 9, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 6, 3, 1, 1, 1, 5, 10, 1, 1, 3, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Jan 29 2023

Keywords

• nonn
• easy
• mult

Comments

The number of unitary divisors of n that are squares is A056624(n) and their sum is A358347(n).

The unitary analog of A069290.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001620, A056624, A069290, A306016, A358347.

Programs

Mathematica

f[p_, e_] := If[OddQ[e], 1, p^(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]%2, 1, f[i, 1]^(f[i, 2]/2) + 1)); }

Formula

a(n) = Sum_{d|n, gcd(d, n/d)=1, d square} sqrt(d).

Multiplicative with a(p^e) = p^(e/2) + 1 if e is even, and 1 if e is odd.

Dirichlet g.f.: zeta(s)*zeta(2*s-1)/zeta(3*s-1).

Sum_{k=1..n} a(k) ~ (3*n/Pi^2)*(log(n) + 3*gamma - 1 - 3*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).