A209802 Partial sums of exponential Möbius function, A166234.

1, 2, 3, 2, 3, 4, 5, 4, 3, 4, 5, 4, 5, 6, 7, 7, 8, 7, 8, 7, 8, 9, 10, 9, 8, 9, 8, 7, 8, 9, 10, 9, 10, 11, 12, 13, 14, 15, 16, 15, 16, 17, 18, 17, 16, 17, 18, 18, 17, 16, 17, 16, 17, 16, 17, 16, 17, 18, 19, 18, 19, 20, 19, 20, 21, 22, 23, 22, 23, 24, 25, 26

Offset: 1

Author

Reinhard Zumkeller, Mar 13 2012

Keywords

• nonn

Comments

Analog of Mertens's function, A002321; conjecture: a(n) > 0.

Values of a(10^n) at n = 1, 2, 3, ...: 4, 34, 355, 3610, 36116, 360967, 3609566, 36094237, .... - Charles R Greathouse IV, Sep 02 2015

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• László Tóth, On certain arithmetic functions involving exponential divisors, II, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166; arXiv preprint, arXiv:0708.3557 [math.NT], 2007-2009.

Crossrefs

Cf. A002321, A166234.

Programs

Haskell

a209802 n = a209802_list !! (n-1)
a209802_list = scanl1 (+) a166234_list

Mathematica

f[p_, e_] := MoebiusMu[e]; em[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate @ Array[em, 100] (* Amiram Eldar, Nov 08 2020 *)

PARI

first(n)=my(s); vector(n,k, s+=factorback(apply(moebius, factor(k)[,2]))) \\ Charles R Greathouse IV, Sep 02 2015

PARI

a(n)=sum(k=1,n,factorback(apply(moebius, factor(k)[,2]))) \\ Charles R Greathouse IV, Sep 02 2015

Formula

a(n) ~ c * n, where c = Product_{p prime} (1 + Sum_{k>=2} (mu(k) - mu(k-1))/p^k) = 0.3609447238... (Tóth, 2007). - Amiram Eldar, Nov 08 2020