This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
t = 0; v = []; for( n = 1, 80, t1 = t; t = t + moebius( n) / n; if( t * t1 < 0, v = concat( v, denominator( t)), )); v
a(6) = 15 because 1 - 1/2 - 1/3 - 1/5 + 1/6 = 4/30 = 2/15.
Table[ Denominator[ Sum[ MoebiusMu[k]/k, {k, 1, n}]], {n, 1, 37}]
t = 0; v = []; for( n = 1, 30, t = t + moebius( n) / n; v = concat( v, denominator( t))); v
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
n = 25; Accumulate[Table[MoebiusMu[k]/k, {k, 1, n}]] * Range[n]! (* Amiram Eldar, Oct 22 2020 *)
from math import factorial
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 A068337(n): return factorial(n)*f(n) # Chai Wah Wu, Nov 03 2023
a(1)= -1 because numerator first sign change is 1-1/2-1/3-1/5= > (-1)/30 <0. a(2)= 2 because next sign change is -1/30+1/6 = 2/15, reverts to positive.
t = 0; v = []; for( n = 1, 120, t1 = t; t = t + moebius(n) / n; if( t * t1 < 0, v = concat( v, numerator( t)), )); v
s = 1; t = 0; Do[t = t + MoebiusMu[n] / n; If[ s > Abs[t], s = Abs[t]; Print[n]], {n, 1, 2 * 10^7}]
t = 0.; t1 = 1; v = []; for( n = 1, 1400, t = t + moebius( n) / n; if( (t / t1 )^2 < 1, t1 = t; v = concat( v, n), )); v