A079057 a(n) = Sum_{k=1..n} bigomega(tau(k)).

0, 1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 20, 21, 22, 24, 25, 27, 29, 31, 32, 35, 36, 38, 40, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 60, 62, 65, 66, 69, 70, 72, 74, 76, 77, 79, 80, 82, 84, 86, 87, 90, 92, 95, 97, 99, 100, 103, 104, 106, 108, 109, 111, 114, 115, 117, 119

Offset: 1

Author

Benoit Cloitre, Feb 02 2003

Keywords

• nonn
• easy

References

• József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter V, page 164.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• E. Heppner, Über die Iteration von Teilerfunktionen, Journal für die reine und angewandte Mathematik, Vol. 265 (1974), pp. 176-182.
• G. J. Rieger, Über einige arithmetische Summen, Manuscripta Mathematica, Vol. 7 (1972), pp. 23-34.

Crossrefs

Partial sums of A058061.

Cf. A000005, A001222, A076191, A077761.

Programs

Mathematica

Accumulate[PrimeOmega[DivisorSigma[0,Range[70]]]] (* Harvey P. Dale, Dec 05 2013 *)

PARI

a(n)=sum(i=1,n,bigomega(numdiv(i)))

Formula

a(n) = n*log(log(n)) + O(n).

a(n) = b * n * log(log(n)) + Sum_{k=0..floor(sqrt(n))} b_k * n/log(n)^k + O(n * exp(-c*sqrt(log(n)))), where b, b_k and c are constants (Heppner, 1974). b = 1 and b_0 = B + C, where B is Mertens's constant (A077761), C = Sum_{k>=2} A076191(k)*P(k) = 0.12861146810484151346..., and P(s) is the prime zeta function. - Amiram Eldar, Jan 15 2024 and Feb 11 2024