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

Author

Reinhard Zumkeller, Feb 12 2012

Keywords

• nonn
• mult
• easy

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.

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.

Programs

Haskell

a206787 = sum . filter odd . a206778_row

Magma

[&+[d:d in Divisors(m)|IsOdd(d) and IsSquarefree(d)]:m in [1..72]]; // Marius A. Burtea, Aug 14 2019

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 *)

PARI

a(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d)); \\ Michel Marcus, Sep 21 2014

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

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