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

1,2

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.

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

Mathematica

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

Prog

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

Crossrefs

Cf. A002321, A166234.

Sequence in context: A125929 A309236 A071933 * A064672 A138554 A063772

Adjacent sequences: A209799 A209800 A209801 * A209803 A209804 A209805

Keyword

nonn

Author

Reinhard Zumkeller, Mar 13 2012

Status

approved