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A352047 Sum of the divisor complements of the odd proper divisors of n. 11
0, 2, 3, 4, 5, 8, 7, 8, 12, 12, 11, 16, 13, 16, 23, 16, 17, 26, 19, 24, 31, 24, 23, 32, 30, 28, 39, 32, 29, 48, 31, 32, 47, 36, 47, 52, 37, 40, 55, 48, 41, 64, 43, 48, 77, 48, 47, 64, 56, 62, 71, 56, 53, 80, 71, 64, 79, 60, 59, 96, 61, 64, 103, 64, 83, 96, 67, 72, 95, 96, 71, 104 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
FORMULA
a(n) = n * Sum_{d|n, d<n, d odd} 1 / d.
a(2n+1) = A000593(2n+1) - 1. - Chai Wah Wu, Mar 01 2022
G.f.: Sum_{k>=2} k * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 14 2023
exp( 2*Sum_{n>=1} a(n)*x^n/n ) is the g.f. of A300415. - Paul D. Hanna, May 15 2023
From Amiram Eldar, Oct 13 2023: (Start)
a(n) = A000593(n) * A006519(n) - A000035(n).
Sum_{k=1..n} a(k) = c * n^2 / 2, where c = Pi^2/8 = 1.23370055... (A111003). (End)
EXAMPLE
a(10) = 12; a(10) = 10 * Sum_{d|10, d<10, d odd} 1 / d = 10 * (1/1 + 1/5) = 12.
MAPLE
A352047 := proc(n)
local a, d ;
a := 0 ;
for d in numtheory[divisors](n) do
if type(d, 'odd') and d < n then
a := a+n/d ;
end if;
end do:
a ;
end proc:
seq(A352047(n), n=1..30) ; # R. J. Mathar, Mar 09 2022
MATHEMATICA
Table[n DivisorSum[n, 1/# &, # < n && OddQ[#] &], {n, 72}] (* Michael De Vlieger, Mar 02 2022 *)
a[n_] := DivisorSigma[-1, n / 2^IntegerExponent[n, 2]] * n - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)
PROG
(PARI) a(n) = n*sumdiv(n, d, if ((d%2) && (d<n), 1/d)); \\ Michel Marcus, Mar 02 2022
(PARI) a(n) = n * sigma(n >> valuation(n, 2), -1) - n % 2; \\ Amiram Eldar, Oct 13 2023
(Python)
from math import prod
from sympy import factorint
def A352047(n): return prod(p**e if p == 2 else (p**(e+1)-1)//(p-1) for p, e in factorint(n).items()) - n % 2 # Chai Wah Wu, Mar 02 2022
CROSSREFS
Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), this sequence (k=1), A352048 (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).
Sequence in context: A326066 A080785 A319605 * A245822 A357260 A069797
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Mar 01 2022
STATUS
approved

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Last modified March 5 02:31 EST 2024. Contains 370537 sequences. (Running on oeis4.)