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A375081
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).
1
5, 5, 5, 17, 17, 14, 14, 14, 14, 14, 32, 34, 34, 34, 27, 27, 27, 27, 23, 23, 27, 51, 51, 51, 51, 44, 44, 44, 44, 44, 39, 39, 39, 39, 39, 44, 74, 74, 74, 74, 74, 74, 74, 74, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 59, 71, 71, 71, 71, 71, 71, 71, 71, 76, 76, 76
OFFSET
1,1
LINKS
Thomas Bloom, Problem 290
Wouter van Doorn, On the non-monotonicity of the denominator of generalized harmonic sums, arXiv:2411.03073 [math.NT], 2024.
FORMULA
a(n) < 4.374*n for all n > 1. - Wouter van Doorn, Nov 06 2024
EXAMPLE
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.
PROG
(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))
(Python)
from fractions import Fraction
from itertools import count
def A375081(n):
a = Fraction((n<<1)+1, n*(n+1))
for k in count(n+1):
if a.denominator > (a:=a+Fraction(1, k+1)).denominator:
return k # Chai Wah Wu, Jul 30 2024
CROSSREFS
Sequence in context: A183390 A013607 A260960 * A283711 A365368 A098526
KEYWORD
nonn
AUTHOR
Ralf Stephan, Jul 29 2024
EXTENSIONS
a(56) onwards from Bhavik Mehta, Jul 31 2024
STATUS
approved