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

%I A263086 #28 Oct 23 2023 16:15:32
%S A263086 2,5,9,13,17,23,27,32,38,44,48,56,60,66,74,80,84,93,97,105,113,119,
%T A263086 123,133,139,145,153,161,165,177,181,188,196,202,210,222,226,232,240,
%U A263086 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
%N A263086 Partial sums of A099777, where A099777(n) gives the number of divisors of n-th even number.
%H A263086 Antti Karttunen, <a href="/A263086/b263086.txt">Table of n, a(n) for n = 1..10082</a>
%H A263086 A. Karttunen, <a href="https://oeis.org/plot2a?name1=A263086&amp;name2=A263085&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawlines=true">Ratio a(n)/A263085(n) drawn with OEIS Plot2-script</a>
%F A263086 a(1) = 2; for n > 1, a(n) = A000005(2*n) + a(n-1) [where A000005(k) gives the number of divisors of k].
%F A263086 Other identities. For all n >= 1:
%F A263086 a(n) = A263084(n) + A263085(n).
%F A263086 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
%F A263086 From _Ridouane Oudra_, Aug 24 2019: (Start)
%F A263086 a(n) = Sum_{k=1..n} A000005(2*k)
%F A263086 a(n) = A006218(n) + A060831(n). (End)
%p A263086 with(numtheory): seq(add(tau(2*k), k=1..n), n= 1..60); # _Ridouane Oudra_, Aug 24 2019
%t A263086 Accumulate[DivisorSigma[0, 2 Range@ 69]] (* _Michael De Vlieger_, Oct 13 2015 *)
%o A263086 (Scheme, with memoization-macro definec)
%o A263086 (definec (A263086 n) (if (= 1 n) (A099777 n) (+ (A099777 n) (A263086 (- n 1)))))
%o A263086 (PARI) a(n) = sum(k=1, n, numdiv(2*k)); \\ _Michel Marcus_, Aug 25 2019
%o A263086 (Python)
%o A263086 from math import isqrt
%o A263086 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
%Y A263086 Cf. A000005, A006218, A099777, A263084, A263085, A060831.
%Y A263086 Cf. also A262518, A262519.
%K A263086 nonn
%O A263086 1,1
%A A263086 _Antti Karttunen_, Oct 12 2015