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A068340

a(n) = Sum_{k=1..n} mu(k)*k, where mu(k) is the Moebius function.

1, -1, -4, -4, -9, -3, -10, -10, -10, 0, -11, -11, -24, -10, 5, 5, -12, -12, -31, -31, -10, 12, -11, -11, -11, 15, 15, 15, -14, -44, -75, -75, -42, -8, 27, 27, -10, 28, 67, 67, 26, -16, -59, -59, -59, -13, -60, -60, -60, -60, -9, -9, -62, -62, -7, -7, 50, 108, 49

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OFFSET

1,3

COMMENTS

Row sums of triangle A143158. - Gary W. Adamson, Jul 27 2008

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

FORMULA

G.f. A(x) satisfies x = Sum_{k>=1} k * (1 - x^k) * A(x^k). - Seiichi Manyama, Apr 01 2023

Sum_{k=1..n} k * a(floor(n/k)) = 1. - Seiichi Manyama, Apr 03 2023

MAPLE

with(numtheory):

a:= proc(n) a(n):= mobius(n)*n +a(n-1) end: a(0):=0:

seq(a(n), n=1..100); # Alois P. Heinz, Oct 21 2012

MATHEMATICA

Table[Sum[MoebiusMu[k]k, {k, n}], {n, 60}] (* Harvey P. Dale, Feb 01 2012 *)

PROG

(Haskell)

a068340 n = a068340_list !! (n-1)

a068340_list = scanl1 (+) a055615_list

-- Reinhard Zumkeller, Sep 04 2015

(PARI) a(n) = sum(k=1, n, k*moebius(k)); \\ Michel Marcus, Jan 14 2023

(Python)

from functools import lru_cache

@lru_cache(maxsize=None)

def A068340(n):

if n <= 1:

return 1

c, j = 1, 2

k1 = n//j

while k1 > 1:

j2 = n//k1 + 1

c -= (j2*(j2-1)-j*(j-1)>>1)*A068340(k1)

j, k1 = j2, n//j2

return c-(n*(n+1)-(j-1)*j>>1) # Chai Wah Wu, Apr 04 2023

CROSSREFS

Cf. A002321, A336276, A336277, A336278, A336279.

Cf. A008683, A143158, A055615, A359478, A361982.

Sequence in context: A115365 A263491 A272427 * A097667 A202388 A283919

Adjacent sequences: A068337 A068338 A068339 * A068341 A068342 A068343

KEYWORD

sign

AUTHOR

Leroy Quet, Feb 27 2002

STATUS

approved