VOOZH about

URL: https://oeis.org/A206787

⇱ A206787 - OEIS

login
A206787
Sum of the odd squarefree divisors of n.
18
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
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).
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
From Ridouane Oudra, Mar 18 2026: (Start)
a(n) = Product_{p|n, p odd prime} (p+1).
a(n) = Sum_{k=0..v_2(n)} (-2)^k*A048250(n/2^k), where v_2 = A007814.
a(n) = Sum_{d|n} d*A154269(d)*A048250(n/d).
a(n) = Sum_{d|n} A193356(d)*A008966(d).
a(n) = A001615(A204455(n)).
a(n) = A048250(n)/A176040(n).
a(n) = A347385(n)/A336651(n).
G.f.: Sum_{k>=1} mu(2*k)^2*k*x^k/(1-x^k). (End)
MAPLE
seq(mul(p+1, p in numtheory[factorset](n) minus {2}), n=1..120); # Ridouane Oudra, Mar 18 2026
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
Inverse Möbius transform of the absolute values of A349343.
Sequence in context: A127555 A192085 A117001 * A327419 A192066 A347385
KEYWORD
nonn,mult,easy
AUTHOR
Reinhard Zumkeller, Feb 12 2012
STATUS
approved