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A263086
Partial sums of A099777, where A099777(n) gives the number of divisors of n-th even number.
8
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
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
KEYWORD
nonn
AUTHOR
Antti Karttunen, Oct 12 2015
STATUS
approved