A350146 Partial sums of A002131.

1, 3, 7, 11, 17, 25, 33, 41, 54, 66, 78, 94, 108, 124, 148, 164, 182, 208, 228, 252, 284, 308, 332, 364, 395, 423, 463, 495, 525, 573, 605, 637, 685, 721, 769, 821, 859, 899, 955, 1003, 1045, 1109, 1153, 1201, 1279, 1327, 1375, 1439, 1496, 1558, 1630, 1686, 1740, 1820

Offset: 1

Author

Seiichi Manyama, Dec 16 2021

Keywords

• nonn

Crossrefs

Cf. A002131, A024916, A222068.

Programs

Mathematica

f[2, e_] := 2^e; f[p_, e_] := (p^(e + 1) - 1)/(p - 1); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate @ Array[s, 50] (* Amiram Eldar, Dec 17 2021 *)

PARI

a(n) = sum(k=1, n, sumdiv(k, d, k/d%2*d));

PARI

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1-x^(2*k)))/(1-x))

Python

def A350146(n): return sum(k*(n//k) for k in range(1,n+1))-sum(k*(n//2//k) for k in range(1,n//2+1)) # Chai Wah Wu, Dec 17 2021

Python

from math import isqrt
def A350146(n): return (-(s:=isqrt(n))**2*(s+1) + sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))+(t:=isqrt(m:=n>>1))**2*(t+1) - sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1)))>>1 # Chai Wah Wu, Oct 21 2023

Formula

a(n) = Sum_{k=1..n} Sum_{d|k, k/d odd} d = Sum_{k=1..n} A002131(k).

G.f.: (1/(1 - x)) * Sum_{k>=1} k * x^k/(1 - x^(2*k)).

a(n) ~ (Pi^2/16) * n^2. - Amiram Eldar, Dec 17 2021

a(n) = A024916(n) - A024916(floor(n/2)). - Chai Wah Wu, Dec 17 2021