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

1,9

Comments

First differs from A336649 at n = 27.

Links

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

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).

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]

Prog

(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))); }

Crossrefs

Cf. A001620, A069290, A336649.

Sequence in context: A368169 A367418 A360164 * A336649 A365487 A368172

Adjacent sequences: A360160 A360161 A360162 * A360164 A360165 A360166

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 29 2023

Status

approved