A375081 - OEIS

%I #37 Feb 14 2025 16:04:26

%S 5,5,5,17,17,14,14,14,14,14,32,34,34,34,27,27,27,27,23,23,27,51,51,51,

%T 51,44,44,44,44,44,39,39,39,39,39,44,74,74,74,74,74,74,74,74,65,65,65,

%U 65,65,65,65,65,65,65,59,71,71,71,71,71,71,71,71,76,76,76

%N Smallest k>n such that the denominator of Sum {i=n..k} (1/i) is larger than the denominator of Sum {i=n..k+1} (1/i).

%H Bhavik Mehta, <a href="/A375081/b375081.txt">Table of n, a(n) for n = 1..10000</a>

%H Thomas Bloom, <a href="https://www.erdosproblems.com/290">Problem 290</a>

%H Wouter van Doorn, <a href="https://arxiv.org/abs/2411.03073">On the non-monotonicity of the denominator of generalized harmonic sums</a>, arXiv:2411.03073 [math.NT], 2024.

%F a(n) < 4.374*n for all n > 1. - _Wouter van Doorn_, Nov 06 2024

%F a(n) > n + 0.54*log(n) for all large enough n, and there are infinitely many n with a(n) < n + 0.61*log(n). - _Wouter van Doorn_, Feb 06 2025

%e 1/3+1/4+1/5=47/60 and 1/3+1/4+1/5+1/6=19/20, and 60>20, so a(3)=5.

%o (PARI) a(n) = for(k=0, oo, my(s=sum(n=n, n+k, 1/n)); if(denominator(s)>denominator(s+1/(n+k+1)), return(n+k); break))

%o (Python)

%o from fractions import Fraction

%o from itertools import count

%o def A375081(n):

%o     a = Fraction((n<<1)+1,n*(n+1))

%o     for k in count(n+1):

%o         if a.denominator > (a:=a+Fraction(1,k+1)).denominator:

%o             return k # _Chai Wah Wu_, Jul 30 2024

%K nonn

%O 1,1

%A _Ralf Stephan_, Jul 29 2024

%E a(56) onwards from _Bhavik Mehta_, Jul 31 2024