A052488 a(n) = floor(n*H(n)) where H(n) is the n-th harmonic number, Sum_{k=1..n} 1/k (A001008/A002805).

1, 3, 5, 8, 11, 14, 18, 21, 25, 29, 33, 37, 41, 45, 49, 54, 58, 62, 67, 71, 76, 81, 85, 90, 95, 100, 105, 109, 114, 119, 124, 129, 134, 140, 145, 150, 155, 160, 165, 171, 176, 181, 187, 192, 197, 203, 208, 214, 219, 224, 230, 235, 241, 247, 252, 258, 263, 269

Offset: 1

Author

Tomas Mario Kalmar (TomKalmar(AT)aol.com), Mar 15 2000

Keywords

• easy
• nonn

Comments

Floor(n*H(n)) gives a (very) rough approximation to the n-th prime.

a(n) is the integer part of the solution to the Coupon Collector's Problem. For example, if there are n=4 different prizes to collect from cereal boxes and they are equally likely to be found, then the integer part of the average number of boxes to buy before the collection is complete is a(4)=8. - Ron Lalonde (ronronronlalonde(AT)hotmail.com), Feb 04 2004

References

• John D. Barrow, One Hundred Essential Things You Didn't Know You Didn't Know, Ch. 3, 'On the Cards', W. W. Norton & Co., NY & London, 2008, pp. 30-32.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001008, A002805, A006218, A060293.

Cf. A001620, A073004.

Programs

Magma

[Floor(n*HarmonicNumber(n)): n in [1..60]]; // G. C. Greubel, May 14 2019

Maple

for n from 1 to 100 do printf(`%d,`,floor(n*sum(1/k, k=1..n))) od:
# Alternatively:
A052488:= n -> floor(n*(Psi(n+1)+gamma));
seq(A052488(n),n=1..100); # Robert Israel, May 19 2014

Mathematica

f[n_] := Floor[n*HarmonicNumber[n]]; Array[f, 60] (* Robert G. Wilson v, Nov 23 2015 *)

PARI

a(n) = floor(n*sum(k=1, n, 1/k)) \\ Altug Alkan, Nov 23 2015

Python

from math import floor
n=100 #number of terms
ans=0
finalans = []
for i in range(1, n+1):
    ans+=(1/i)
    finalans.append(floor(ans*i))
print(finalans)
# Adam Hugill, Feb 14 2022

Python

from fractions import Fraction
from itertools import count, islice
def agen():
    Hn = 0
    for n in count(1):
        Hn += Fraction(1, n)
        yield (n*Hn.numerator)//Hn.denominator
print(list(islice(agen(), 60))) # Michael S. Branicky, Aug 10 2022

Python

from sympy import harmonic
def A052488(n): return int(n*harmonic(n)) # Chai Wah Wu, Oct 24 2023

Sage

[floor(n*harmonic_number(n)) for n in (1..60)] # G. C. Greubel, May 14 2019

Extensions

More terms from James Sellers, Mar 17 2000