A326124 a(n) is the sum of all divisors of the first n positive even numbers.

3, 10, 22, 37, 55, 83, 107, 138, 177, 219, 255, 315, 357, 413, 485, 548, 602, 693, 753, 843, 939, 1023, 1095, 1219, 1312, 1410, 1530, 1650, 1740, 1908, 2004, 2131, 2275, 2401, 2545, 2740, 2854, 2994, 3162, 3348, 3474, 3698, 3830, 4010, 4244, 4412, 4556, 4808, 4979, 5196, 5412, 5622, 5784, 6064, 6280

Offset

1,1

Comments

A326123(n)/a(n) converges to 3/5.

a(n) is also the total area of the terraces of the first n even-indexed levels of the stepped pyramid described in A245092.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = A024916(2n) - A326123(n).

a(n) ~ 5 * Pi^2 * n^2 / 24. - Vaclav Kotesovec, Aug 18 2021

Example

For n = 3 the first three positive even numbers are [2, 4, 6] and their divisors are [1, 2], [1, 2, 4], [1, 2, 3, 6] respectively, and the sum of these divisors is 1 + 2 + 1 + 2 + 4 + 1 + 2 + 3 + 6 = 22, so a(3) = 22.

Maple

ListTools:-PartialSums(map(numtheory:-sigma, [seq(i, i=2..200, 2)])); # Robert Israel, Jun 12 2019

Mathematica

Accumulate@ DivisorSigma[1, Range[2, 110, 2]] (* Michael De Vlieger, Jun 09 2019 *)

Prog

(PARI) terms(n) = my(s=0, i=0); for(k=1, n-1, if(i>=n, break); s+=sigma(2*k); print1(s, ", "); i++)
/* Print initial 50 terms as follows: */
terms(50) \\ Felix Fröhlich, Jun 08 2019
(PARI) a(n) = sum(k=1, 2*n, if (!(k%2), sigma(k))); \\ Michel Marcus, Jun 08 2019
(Python)
from math import isqrt
def A326124(n): return (t:=isqrt(m:=n>>1))**2*(t+1) - sum((q:=m//k)*((k<<1)+q+1) for k in range(1, t+1))-3*((s:=isqrt(n))**2*(s+1) - sum((q:=n//k)*((k<<1)+q+1) for k in range(1, s+1))>>1) # Chai Wah Wu, Oct 21 2023

Crossrefs

Partial sums of A062731.

Cf. A000203, A005843, A024916, A237593, A245092, A299174, A326123.

Sequence in context: A161672 A190092 A174459 * A346166 A122795 A140066

Adjacent sequences: A326121 A326122 A326123 * A326125 A326126 A326127

Keyword

nonn,easy

Author

Omar E. Pol, Jun 07 2019

Status

approved