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

1,2

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

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 *)

Prog

(PARI) a(n) = floor(n*sum(k=1, n, 1/k)) \\ Altug Alkan, Nov 23 2015
(Magma) [Floor(n*HarmonicNumber(n)): n in [1..60]]; // G. C. Greubel, May 14 2019
(Sage) [floor(n*harmonic_number(n)) for n in (1..60)] # G. C. Greubel, May 14 2019
(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

Crossrefs

Cf. A001008, A002805, A006218, A060293.

Cf. A001620, A073004.

Sequence in context: A310027 A310028 A287414 * A076372 A248611 A005356

Adjacent sequences: A052485 A052486 A052487 * A052489 A052490 A052491

Keyword

easy,nonn

Author

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

Extensions

More terms from James A. Sellers, Mar 17 2000

Status

approved