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

1,2

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

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.

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

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

Sequence in context: A257064 A085800 A155190 * A153776 A005625 A340989

Adjacent sequences: A333314 A333315 A333316 * A333318 A333319 A333320

Keyword

nonn

Author

Amiram Eldar, Mar 14 2020

Status

approved