A326123 a(n) is the sum of all divisors of the first n odd numbers.

1, 5, 11, 19, 32, 44, 58, 82, 100, 120, 152, 176, 207, 247, 277, 309, 357, 405, 443, 499, 541, 585, 663, 711, 768, 840, 894, 966, 1046, 1106, 1168, 1272, 1356, 1424, 1520, 1592, 1666, 1790, 1886, 1966, 2087, 2171, 2279, 2399, 2489, 2601, 2729, 2849, 2947, 3103, 3205, 3309, 3501, 3609, 3719

Offset

1,2

Comments

a(n)/A326124(n) converges to 3/5.

a(n) is also the total area of the terraces of the first n odd-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) - A326124(n).

a(n) ~ Pi^2 * n^2 / 8. - Vaclav Kotesovec, Aug 18 2021

Example

For n = 3 the first three odd numbers are [1, 3, 5] and their divisors are [1], [1, 3], [1, 5] respectively, and the sum of these divisors is 1 + 1 + 3 + 1 + 5 = 11, so a(3) = 11.

Maple

ListTools:-PartialSums(map(numtheory:-sigma, [seq(i, i=1..200, 2)])); # Robert Israel, Jun 12 2019

Mathematica

Accumulate@ DivisorSigma[1, Range[1, 109, 2]] (* Michael De Vlieger, Jun 09 2019 *)

Prog

(PARI) terms(n) = my(s=0, i=0); for(k=0, n-1, if(i>=n, break); s+=sigma(2*k+1); print1(s, ", "); i++)
/* Print initial 50 terms as follows: */
terms(50) \\ Felix Fröhlich, Jun 08 2019
(PARI) a(n) = sum(k=1, 2*n-1, if (k%2, sigma(k))); \\ Michel Marcus, Jun 08 2019
(Python)
from math import isqrt
def A326123(n): return (-(s:=isqrt(r:=n<<1))**2*(s+1) + sum((q:=r//k)*((k<<1)+q+1) for k in range(1, s+1))>>1) -(t:=isqrt(m:=n>>1))**2*(t+1)+sum((q:=m//k)*((k<<1)+q+1) for k in range(1, t+1))+3*((u:=isqrt(n))**2*(u+1)-sum((q:=n//k)*((k<<1)+q+1) for k in range(1, u+1))>>1) # Chai Wah Wu, Nov 01 2023

Crossrefs

Partial sums of A008438.

Cf. A000203, A005408, A024916, A237593, A245092, A326124.

Sequence in context: A023245 A125003 A062718 * A288112 A225250 A105914

Adjacent sequences: A326120 A326121 A326122 * A326124 A326125 A326126

Keyword

nonn,easy

Author

Omar E. Pol, Jun 07 2019

Status

approved