A360163 a(n) is the sum of the square roots of the divisors of n that are odd squares.

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 6, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 8, 6, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 6, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Jan 29 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A336649 at n = 27.

Links

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

Crossrefs

Cf. A001620, A069290, A336649.

Programs

Mathematica

f[p_, e_] := (p^(Floor[e/2] + 1) - 1)/(p - 1); f[2, e_] := 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, 1] == 2, 1, (f[i, 1]^(floor(f[i, 2]/2)+1) - 1)/(f[i, 1] - 1))); }

Formula

a(n) = Sum_{d|n, d odd square} sqrt(d).

a(n) = (A069290(n) + A347176(n))/2.

a(n) = A069290(n) if n is not a multiple of 4.

Multiplicative with a(2^e) = 1, and a(p^e) = (p^(floor(e/2)+1)-1)/(p-1) for p > 2.

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

Sum_{k=1..n} a(k) ~ (n/4) * (log(n) + 3*gamma - 1 + 2*log(2)), where gamma is Euler's constant (A001620).