A357844 Denominators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).
1, 2, 1, 3, 6, 12, 12, 6, 2, 4, 4, 12, 12, 6, 12, 60, 60, 60, 60, 20, 5, 20, 20, 40, 120, 120, 120, 120, 120, 15, 30, 5, 20, 5, 20, 180, 180, 90, 180, 360, 360, 45, 90, 45, 90, 180, 180, 180, 180, 180, 45, 90, 45, 360, 360, 45, 180, 45, 90, 180, 180, 45, 90, 630
Offset: 1
Links
- László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.
Programs
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Mathematica
Denominator[Accumulate[Array[(-1)^(# + 1)/DivisorSigma[0, #] &, 60]]]
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PARI
lista(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / numdiv(k); print1(denominator(s), ", "))}; -
Python
from fractions import Fraction from sympy import divisor_count def A357844(n): return sum(Fraction(1 if k&1 else -1, divisor_count(k)) for k in range(1,n+1)).denominator # Chai Wah Wu, Oct 16 2022
Formula
a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/d(k)), where d(k) = A000005(k).
Comments