OFFSET
1,2
COMMENTS
The value of (1/n)*Sum_{k=1..n} sigma(k)/k approaches Pi^2/6.
REFERENCES
József Sándor, Dragoslav S. Mitrinović, and Borislav Crstici, Handbook of Number Theory I, Springer, 2006, Section III.5, p. 82.
Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Deutscher Verlag der Wissenschaften, Berlin, 1963, p. 99.
LINKS
FORMULA
G.f.: (1/(1 - x))*Sum_{k>=1} log(1/(1 - x^k)) (for a(n)/A284650(n), see example).
a(n) = numerator of Sum_{k=1..n} Sum_{d|k} 1/d.
a(n) = numerator of Sum_{k=1..n} sigma(k)/k.
a(n) = numerator of Sum_{k=1..n} floor(n/k)/k. - Ridouane Oudra, Jan 21 2024
EXAMPLE
1, 5/2, 23/6, 67/12, 407/60, 527/60, 4169/420, 9913/840, 33379/2520, 7583/504, 89461/5544, 102397/5544, 1408777/72072, 1532329/72072, 8238221/360360, ...
MAPLE
with(numtheory): seq(numer(add(sigma(k)/k, k=1..n)), n=1..40); # Ridouane Oudra, Jan 21 2024
MATHEMATICA
Table[Numerator[Sum[DivisorSigma[-1, k], {k, 1, n}]], {n, 1, 29}]
(* Alternative: *)
Table[Numerator[Sum[DivisorSigma[1, k]/k, {k, 1, n}]], {n, 1, 29}]
(* Alternative: *)
nmax = 29; Rest[Numerator[CoefficientList[Series[1/(1 - x) Sum[Log[1/(1 - x^k)], {k, 1, nmax}], {x, 0, nmax}], x]]]
PROG
(PARI) for(n=1, 29, print1(numerator(sum(k=1, n, sigma(k)/k)), ", ")) \\ Indranil Ghosh, Mar 31 2017
(Python)
from sympy import divisor_sigma, Integer
print([sum(divisor_sigma(k)/Integer(k) for k in range(1, n + 1)).numerator for n in range(1, 30)]) # Indranil Ghosh, Mar 31 2017
(Python)
from fractions import Fraction
from sympy import harmonic
def A284648(n):
c, j, v = Fraction(0), 1, 0
while j <= n:
k = n//j
m = n//k
c += k*(-v+(v:=harmonic(m)))
j = m+1
return c.p # Chai Wah Wu, May 19 2026
CROSSREFS
KEYWORD
nonn,frac,changed
AUTHOR
Ilya Gutkovskiy, Mar 31 2017
STATUS
approved
