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A333317
Partial sums of A248577.
1
0, 2, 4, 7, 9, 17, 19, 23, 26, 34, 36, 48, 50, 58, 66, 71, 73, 85, 87, 99, 107, 115, 117, 133, 136, 144, 148, 160, 162, 186, 188, 194, 202, 210, 218, 236, 238, 246, 254, 270, 272, 296, 298, 310, 322, 330, 332, 352, 355, 367, 375, 387, 389, 405, 413, 429, 437
OFFSET
1,2
REFERENCES
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.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 161.
LINKS
Jean-Marie De Koninck and Armel Mercier, Remarque Sur un Article de T. M. Apostol, Canadian Mathematical Bulletin, Vol. 20 (1977), pp. 77-88.
Randell Heyman, A summation of the number of distinct prime divisors of the lcm, arXiv:2012.11837 [math.NT], 2020.
FORMULA
a(n) = Sum_{k=1..n} A248577(k) = Sum_{k=1..n} A000005(k) * A001221(k).
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).
Empirical: a(n) = Sum_{i*j <= n} omega(lcm(i, j)). See Heyman. - Michel Marcus, Dec 26 2020
MATHEMATICA
f[n_] := DivisorSigma[0, n] * PrimeNu[n]; Accumulate @ Array[f, 100]
PROG
(PARI) a(n) = sum(k=1, n, numdiv(k)*omega(k)); \\ Michel Marcus, Dec 22 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Mar 14 2020
STATUS
approved