A352047 Sum of the divisor complements of the odd proper divisors of n.

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

Offset

1,2

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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

Cf. A000035, A000593, A002131, A006519, A111003, A300415.

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

Adjacent sequences: A352044 A352045 A352046 * A352048 A352049 A352050

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 01 2022

Status

approved