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A326123 a(n) is the sum of all divisors of the first n odd numbers. 9
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 (list; graph; refs; listen; history; text; internal format)
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
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.
Sequence in context: A023245 A125003 A062718 * A288112 A225250 A105914
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Jun 07 2019
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)