A206787 Sum of the odd squarefree divisors of n.

1, 1, 4, 1, 6, 4, 8, 1, 4, 6, 12, 4, 14, 8, 24, 1, 18, 4, 20, 6, 32, 12, 24, 4, 6, 14, 4, 8, 30, 24, 32, 1, 48, 18, 48, 4, 38, 20, 56, 6, 42, 32, 44, 12, 24, 24, 48, 4, 8, 6, 72, 14, 54, 4, 72, 8, 80, 30, 60, 24, 62, 32, 32, 1, 84, 48, 68, 18, 96, 48, 72, 4

Offset

1,3

Comments

a(A000079(n)) = 1; a(A057716(n)) > 1; a(A065119(n)) = 4; a(A086761(n)) = 6.

Inverse Mobius transform of 1, 0, 3, 0, 5, 0, 7, 0, 0, 0, 11, 0, 13, 0, 15, 0, 17, 0, 19, 0, 21, 0, 23, 0, 0, 0, 0, 0, 29... - R. J. Mathar, Jul 12 2012

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Jon Maiga, Computer-generated formulas for A206787, Sequence Machine.

Formula

a(n) = Sum_{k = 1..A034444(n)} (A206778(n,k) mod 2) * A206778(n,k).

a(n) = Sum_{d|n} d*mu(2*d)^2, where mu is the Möbius function (A008683). - Ridouane Oudra, Aug 14 2019

Multiplicative with a(2^e) = 1, and a(p^e) = p + 1 for p > 2. - Amiram Eldar, Sep 18 2020

Sum_{k=1..n} a(k) ~ (1/3) * n^2. - Amiram Eldar, Nov 17 2022

Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s-2))*(2^s/(2^s+2)). - Amiram Eldar, Jan 03 2023

From Antti Karttunen, Nov 22 2023: (Start)

a(n) = A000203(A204455(n)) = A000593(A007947(n)) = A048250(n)/A010684(n-1). [From Sequence Machine]

a(n) = Sum_{d|n} abs(A349343(d)). [See R. J. Mathar's Jul 12 2012 comment above]

(End)

a(n) = Sum_{d divides n, d odd} d * mu(d)^2. - Peter Bala, Feb 01 2024

Maple

seq(add(d*mobius(2*d)^2, d in divisors(n)), n=1 .. 80); # Ridouane Oudra, Aug 14 2019

Mathematica

a[n_] := DivisorSum[n, #*Boole[OddQ[#] && SquareFreeQ[#]]&]; Array[a, 80] (* Jean-François Alcover, Dec 05 2015 *)
f[2, e_] := 1; f[p_, e_] := p + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

Prog

(Haskell)
a206787 = sum . filter odd . a206778_row
(PARI) a(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d)); \\ Michel Marcus, Sep 21 2014
(Magma) [&+[d:d in Divisors(m)|IsOdd(d) and IsSquarefree(d)]:m in [1..72]]; // Marius A. Burtea, Aug 14 2019
(Python)
from math import prod
from sympy import primefactors
def A206787(n): return prod(1+(p if p>2 else 0) for p in primefactors(n)) # Chai Wah Wu, Oct 10 2024

Crossrefs

Cf. A000203, A000593, A007947, A008683, A056911, A048250, A010684, A192066, A204455.

Cf. A000079, A057716, A065119, A086761.

Inverse Möbius transform of the absolute values of A349343.

Sequence in context: A127555 A192085 A117001 * A327419 A192066 A347385

Adjacent sequences: A206784 A206785 A206786 * A206788 A206789 A206790

Keyword

nonn,mult,easy

Author

Reinhard Zumkeller, Feb 12 2012

Status

approved