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

1,4

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

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).

Mathematica

f[p_, e_] := If[OddQ[e], 1, p^(e/2) + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(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)); }

Crossrefs

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

Sequence in context: A016465 A122947 A367990 * A367988 A363925 A231147

Adjacent sequences: A360159 A360160 A360161 * A360163 A360164 A360165

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 29 2023

Status

approved