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

1,1

Links

Antti Karttunen, Table of n, a(n) for n = 1..10082

A. Karttunen, Ratio a(n)/A263085(n) drawn with OEIS Plot2-script

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)

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 *)

Prog

(Scheme, with memoization-macro definec)
(definec (A263086 n) (if (= 1 n) (A099777 n) (+ (A099777 n) (A263086 (- n 1)))))
(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

Crossrefs

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

Cf. also A262518, A262519.

Sequence in context: A063203 A130244 A161569 * A353036 A182814 A351628

Adjacent sequences: A263083 A263084 A263085 * A263087 A263088 A263089

Keyword

nonn

Author

Antti Karttunen, Oct 12 2015

Status

approved