A070889 Denominator of Sum_{k=1..n} mu(k)/k.

1, 2, 6, 6, 30, 15, 105, 105, 105, 210, 2310, 2310, 30030, 15015, 5005, 5005, 85085, 85085, 1616615, 1616615, 4849845, 9699690, 223092870, 223092870, 223092870, 111546435, 111546435, 111546435, 3234846615, 2156564410, 66853496710

Offset: 1

Author

Donald S. McDonald, May 17 2002

Keywords

• frac
• nonn

Examples

			a(6) = 15 because 1 - 1/2 - 1/3 - 1/5 + 1/6 = 4/30 = 2/15.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..2370

Crossrefs

Cf. A008683, A068337, A070888 (numerators).

Programs

Mathematica

Table[ Denominator[ Sum[ MoebiusMu[k]/k, {k, 1, n}]], {n, 1, 37}]

PARI

t = 0; v = []; for( n = 1, 30, t = t + moebius( n) / n; v = concat( v, denominator( t))); v

Python

from functools import lru_cache
from sympy import harmonic
@lru_cache(maxsize=None)
def f(n):
    if n <= 1:
        return 1
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (harmonic(j-1)-harmonic(j2-1))*f(k1)
        j, k1 = j2, n//j2
    return c+harmonic(j-1)-harmonic(n)
def A070889(n): return f(n).denominator # Chai Wah Wu, Nov 03 2023

Extensions

Edited by Robert G. Wilson v, Jun 10 2002