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

Author

Omar E. Pol, Jun 07 2019

Keywords

• nonn
• easy

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.

Examples

			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.
		

Links

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

Crossrefs

Partial sums of A008438.

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

Programs

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 *)

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

Formula

a(n) = A024916(2n) - A326124(n).

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