A263086 Partial sums of A099777, where A099777(n) gives the number of divisors of n-th even number.

2, 5, 9, 13, 17, 23, 27, 32, 38, 44, 48, 56, 60, 66, 74, 80, 84, 93, 97, 105, 113, 119, 123, 133, 139, 145, 153, 161, 165, 177, 181, 188, 196, 202, 210, 222, 226, 232, 240, 250, 254, 266, 270, 278, 290, 296, 300, 312, 318, 327, 335, 343, 347, 359, 367, 377, 385, 391, 395, 411, 415, 421, 433, 441, 449, 461, 465, 473, 481

Offset: 1

Author

Antti Karttunen, Oct 12 2015

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10082
• A. Karttunen, Ratio a(n)/A263085(n) drawn with OEIS Plot2-script

Crossrefs

Cf. A000005, A006218, A099777, A263084, A263085, A060831.

Cf. also A262518, A262519.

Programs

Maple

with(numtheory): seq(add(tau(2*k), k=1..n), n= 1..60); # Ridouane Oudra, Aug 24 2019

Mathematica

Accumulate[DivisorSigma[0, 2 Range@ 69]] (* Michael De Vlieger, Oct 13 2015 *)

PARI

a(n) = sum(k=1, n, numdiv(2*k)); \\ Michel Marcus, Aug 25 2019

Python

from math import isqrt
def A263086(n): return (t:=isqrt(m:=n>>1))**2-((s:=isqrt(n))**2<<1)+((sum(n//k for k in range(1,s+1))<<1)-sum(m//k for k in range(1,t+1))<<1) # Chai Wah Wu, Oct 23 2023

Formula

a(1) = 2; for n > 1, a(n) = A000005(2*n) + a(n-1) [where A000005(k) gives the number of divisors of k].

Other identities. For all n >= 1:

a(n) = A263084(n) + A263085(n).

a(n) ~ n/2 * (3*log(n) + log(2) + 6*gamma - 3), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 13 2019

From Ridouane Oudra, Aug 24 2019: (Start)

a(n) = Sum_{k=1..n} A000005(2*k)

a(n) = A006218(n) + A060831(n). (End)