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A333317 Partial sums of A248577.

%I A333317 #22 Dec 27 2020 01:55:42
%S A333317 0,2,4,7,9,17,19,23,26,34,36,48,50,58,66,71,73,85,87,99,107,115,117,
%T A333317 133,136,144,148,160,162,186,188,194,202,210,218,236,238,246,254,270,
%U A333317 272,296,298,310,322,330,332,352,355,367,375,387,389,405,413,429,437
%N A333317 Partial sums of A248577.
%D A333317 Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields, North-Holland, 1980, pp. 233-235.
%D A333317 Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 161.
%H A333317 Amiram Eldar, <a href="/A333317/b333317.txt">Table of n, a(n) for n = 1..10000</a>
%H A333317 Jean-Marie De Koninck and Armel Mercier, <a href="https://doi.org/10.4153/CMB-1977-013-8">Remarque Sur un Article de T. M. Apostol</a>, Canadian Mathematical Bulletin, Vol. 20 (1977), pp. 77-88.
%H A333317 Randell Heyman, <a href="https://arxiv.org/abs/2012.11837">A summation of the number of distinct prime divisors of the lcm</a>, arXiv:2012.11837 [math.NT], 2020.
%F A333317 a(n) = Sum_{k=1..n} A248577(k) = Sum_{k=1..n} A000005(k) * A001221(k).
%F A333317 a(n) ~ 2 * n * log(n) * log(log(n)) + 2 * B * n * log(n), where B = M - 1 - S/2  = -0.9646264971..., M is Mertens's constant (A077761) and S = Sum_{p prime} 1/p^2 (A085548).
%F A333317 Empirical: a(n) = Sum_{i*j <= n} omega(lcm(i, j)). See Heyman. - _Michel Marcus_, Dec 26 2020
%t A333317 f[n_] := DivisorSigma[0, n] * PrimeNu[n]; Accumulate @ Array[f, 100]
%o A333317 (PARI) a(n) = sum(k=1, n, numdiv(k)*omega(k)); \\ _Michel Marcus_, Dec 22 2020
%Y A333317 Cf. A000005, A001221, A077761, A085548, A248577.
%K A333317 nonn
%O A333317 1,2
%A A333317 _Amiram Eldar_, Mar 14 2020