A333317 Partial sums of A248577.

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

Author

Amiram Eldar, Mar 14 2020

Keywords

• nonn

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

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• 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.

Crossrefs

Cf. A000005, A001221, A077761, A085548, A248577.

Programs

Mathematica

f[n_] := DivisorSigma[0, n] * PrimeNu[n]; Accumulate @ Array[f, 100]

PARI

a(n) = sum(k=1, n, numdiv(k)*omega(k)); \\ Michel Marcus, Dec 22 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