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

%I A326123 #40 Nov 01 2023 18:30:17
%S A326123 1,5,11,19,32,44,58,82,100,120,152,176,207,247,277,309,357,405,443,
%T A326123 499,541,585,663,711,768,840,894,966,1046,1106,1168,1272,1356,1424,
%U A326123 1520,1592,1666,1790,1886,1966,2087,2171,2279,2399,2489,2601,2729,2849,2947,3103,3205,3309,3501,3609,3719
%N A326123 a(n) is the sum of all divisors of the first n odd numbers.
%C A326123 a(n)/A326124(n) converges to 3/5.
%C A326123 a(n) is also the total area of the terraces of the first n odd-indexed levels of the stepped pyramid described in A245092.
%H A326123 Robert Israel, <a href="/A326123/b326123.txt">Table of n, a(n) for n = 1..10000</a>
%F A326123 a(n) = A024916(2n) - A326124(n).
%F A326123 a(n) ~ Pi^2 * n^2 / 8. - _Vaclav Kotesovec_, Aug 18 2021
%e A326123 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.
%p A326123 ListTools:-PartialSums(map(numtheory:-sigma, [seq(i,i=1..200,2)])); # _Robert Israel_, Jun 12 2019
%t A326123 Accumulate@ DivisorSigma[1, Range[1, 109, 2]] (* _Michael De Vlieger_, Jun 09 2019 *)
%o A326123 (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++)
%o A326123 /* Print initial 50 terms as follows: */
%o A326123 terms(50) \\ _Felix Fröhlich_, Jun 08 2019
%o A326123 (PARI) a(n) = sum(k=1, 2*n-1, if (k%2, sigma(k))); \\ _Michel Marcus_, Jun 08 2019
%o A326123 (Python)
%o A326123 from math import isqrt
%o A326123 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
%Y A326123 Partial sums of A008438.
%Y A326123 Cf. A000203, A005408, A024916, A237593, A245092, A326124.
%K A326123 nonn,easy
%O A326123 1,2
%A A326123 _Omar E. Pol_, Jun 07 2019