A203811 -id:A203811 - cp's OEIS frontend

A002805 Denominators of harmonic numbers H(n) = Sum_{i=1..n} 1/i.

1, 2, 6, 12, 60, 20, 140, 280, 2520, 2520, 27720, 27720, 360360, 360360, 360360, 720720, 12252240, 4084080, 77597520, 15519504, 5173168, 5173168, 118982864, 356948592, 8923714800, 8923714800, 80313433200, 80313433200, 2329089562800, 2329089562800, 72201776446800

Offset: 1

N. J. A. Sloane

H(n)/2 is the maximal distance that a stack of n cards can project beyond the edge of a table without toppling.
If n is not in {1, 2, 6} then a(n) has at least one prime factor other than 2 or 5. E.g., a(5) = 60 has a prime factor 3 and a(7) = 140 has a prime factor 7. This implies that every H(n) = A001008(n)/A002805(n), n not from {1, 2, 6}, has an infinite decimal representation. For a proof see the J. Havil reference. - Wolfdieter Lang, Jun 29 2007
a(n) = A213999(n,n-1). - Reinhard Zumkeller, Jul 03 2012
From Wolfdieter Lang, Apr 16 2015: (Start)
a(n)/A001008(n) = 1/H(n) is the solution of the following version of the classical cistern and pipes problem. A cistern is connected to n different pipes of water. For the k-th pipe it takes k time units (say, days) to fill the empty cistern, for k = 1, 2, ..., n. How long does it take for the n pipes together to fill the empty cistern? 1/H(n) gives the answer as a fraction of the time unit.
In general, if the k-th pipe needs d(k) days to fill the empty cistern then all pipes together need 1/Sum_{k=1..n} 1/d(k) = HM(d(1), ..., d(n))/n days, where HM denotes the harmonic mean HM. For the described problem, HM(1, 2, ..., n)/n = A102928(n)/(n*A175441(n)) = 1/H(n).
For a classical cistern and pipes problem see, e.g., the Hunger-Vogel reference (in Greek and German) given in A256101, problem 27, p. 29, where n = 3, and d(1), d(2) and d(3) are 6, 4 and 3 days. On p. 97 of this reference one finds remarks on the history of such problems (called in German 'Brunnenaufgabe'). (End)
From Wolfdieter Lang, Apr 17 2015: (Start)
An example of the above mentioned cistern and pipes problems appears in Chiu Chang Suan Shu (nine books on arithmetic) in book VI, problem 26. The numbers are there 1/2, 1, 5/2, 3 and 5 (days) and the result is 15/75 (day). See the reference (in German) on p. 68.
A historical account on such cistern problems is found in the Johannes Tropfke reference, given in A256101, section 4.2.1.2 Zisternenprobleme (Leistungsprobleme), pp. 578-579.
In Fibonacci's Liber Abaci such problems appear on p. 281 and p. 284 of L. E. Sigler's translation. (End)
All terms > 1 are even while corresponding numerators (A001008) are all odd (proof in Pólya and Szegő). - Bernard Schott, Dec 24 2021

			H(n) = [ 1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, ... ] = A001008/A002805.

Chiu Chang Suan Shu, Neun Bücher arithmetischer Technik, translated and commented by Kurt Vogel, Ostwalds Klassiker der exakten Wissenschaften, Band 4, Friedr. Vieweg & Sohn, Braunschweig, 1968, p. 68.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 258-261.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 259.
J. Havil, Gamma, (in German), Springer, 2007, p. 35-6; Gamma: Exploring Euler's Constant, Princeton Univ. Press, 2003.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 615.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, volume II, Springer, reprint of the 1976 edition, 1998, problem 251, p. 154.
L. E. Sigler, Fibonacci's Liber Abaci, Springer, 2003, pp. 281, 284.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Kenny Lau, Table of n, a(n) for n = 1..2308 [First 200 terms computed by T. D. Noe]
R. M. Dickau, Harmonic numbers and the book stacking problem.
Haight, Frank A., and Robert B. Jones., "A probabilistic treatment of qualitative data with special reference to word association tests." Journal of Mathematical Psychology 11.3 (1974): 237-244. [Annotated scanned copy]
Frank Haight and N. J. A. Sloane, Correspondence, 1975.
Antal Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
Fredrik Johansson, How (not) to compute harmonic numbers, Feb 21 2009.
Peter Shiu, The denominators of harmonic numbers, arXiv:1607.02863 [math.NT], 2016.
N. J. A. Sloane, Illustration of initial terms.
Jonathan Sondow and Eric W. Weisstein, MathWorld: Harmonic Number.
Eric Weisstein's World of Mathematics, Book Stacking Problem.
Bing-Ling Wu, Yong-Gao Chen, On the denominators of harmonic numbers, arXiv:1711.00184 [math.NT], 2017.

Cf. A001008 (numerators), A075135, A025529, A203810, A203811, A203812.
Partial sums: A027612/A027611 = 1, 5/2, 13/3, 77/12, 87/10, 223/20,...
The following fractions are all related to each other: Sum 1/n: A001008/A002805, Sum 1/prime(n): A024451/A002110 and A106830/A034386, Sum 1/nonprime(n): A282511/A282512, Sum 1/composite(n): A250133/A296358, Sum 1/n^2: A007406/A007407, Sum 1/n^3: A007408/A007409.

List([1..30],n->DenominatorRat(Sum([1..n],i->1/i))); # Muniru A Asiru, Dec 20 2018

import Data.Ratio ((%), denominator)
a002805 = denominator . sum . map (1 %) . enumFromTo 1
a002805_list = map denominator $ scanl1 (+) $ map (1 %) [1..]
-- Reinhard Zumkeller, Jul 03 2012

[Denominator(HarmonicNumber(n)): n in [1..40]]; // Vincenzo Librandi, Apr 16 2015

seq(denom(sum((2*k-1)/k, k=1..n), n=1..30); # Gary Detlefs, Jul 18 2011
f:=n->denom(add(1/k, k=1..n)); # N. J. A. Sloane, Nov 15 2013

Denominator[ Drop[ FoldList[ #1 + 1/#2 &, 0, Range[ 30 ] ], 1 ] ] (* Harvey P. Dale, Feb 09 2000 *)
Table[Denominator[HarmonicNumber[n]], {n, 1, 40}] (* Stefan Steinerberger, Apr 20 2006 *)
Denominator[Accumulate[1/Range[25]]] (* Alonso del Arte, Nov 21 2018 *)

a(n)=denominator(sum(k=2,n,1/k)) \\ Charles R Greathouse IV, Feb 11 2011

from fractions import Fraction
def a(n): return sum(Fraction(1, i) for i in range(1, n+1)).denominator
print([a(n) for n in range(1, 30)]) # Michael S. Branicky, Dec 24 2021

def harmonic(a, b): # See the F. Johansson link.
    if b - a == 1 : return 1, a
    m = (a+b)//2
    p, q = harmonic(a,m)
    r, s = harmonic(m,b)
    return p*s+q*r, q*s
def A002805(n) : H = harmonic(1,n+1); return denominator(H[0]/H[1])
[A002805(n) for n in (1..29)] # Peter Luschny, Sep 01 2012

a(n) = Denominator(Sum_{k=1..n} (2*k-1)/k). - Gary Detlefs, Jul 18 2011
a(n) = n! / gcd(Stirling1(n+1, 2), n!) = n! / gcd(A000254(n),n!). - Max Alekseyev, Mar 01 2018
a(n) = the (reduced) denominator of the continued fraction 1/(1 - 1^2/(3 - 2^2/(5 - 3^2/(7 - ... - (n-1)^2/(2*n-1))))). - Peter Bala, Feb 18 2024

Definition edited by Daniel Forgues, May 19 2010

A203810 Numerators of s(i) = s(i-1) - (1/i)*sign(s(i-1)) with s(1) = 1.

Original entry on oeis.org

1, 1, 1, -1, 7, -1, 13, -9, 199, -53, 1937, -373, 22871, -2869, 4231, -547, 134845, -1291, 2425919, 489967, -10595393, -13913, 232472561, 9379691, -1023947321, 5712079, -2957435363, -89098463, 77729577773, 93259013, -2326198533397, -139786038869, 385098109121

Offset: 1

Hugo Pfoertner and Rainer Rosenthal, Jan 06 2012

Denominators are given in A203811.

Similar to harmonic series, but with signs chosen to minimize the absolute value of the next term.

			s(1)=1, to minimize abs(s(2)) 1/2 has to be subtracted. s(2)=1-1/2=1/2. Similar for s(3) and s(4): s(3)=s(2)-1/3=1/2-1/3=1/6, s(4)=1/6-1/4=-1/12. Since s(4) is negative s(5)=s(4)+1/5=-1/12+1/5=7/60. The numerators of s(1)...s(5) are 1, 1, 1, -1, 7 and the corresponding denominators are 1, 2, 6, 12, 60.

Hugo Pfoertner, Table of n, a(n) for n = 1..200
Hugo Pfoertner, Illustration of A203810(i)/A203811(i) for i<=100
Hugo Pfoertner, Illustration of A203810(i)/A203811(i) for even i, i<=500

Cf. A203811 (denominators), A203812 (minima of abs(A203810(i)/A203811(i))).

Cf. A001008, A002805 (harmonic numbers).

A203812 Numbers n where abs(s(n)) produces a new minimum, with s(1) = 1 and s(i) = s(i-1) - sign(s(i-1))*(1/i).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 22, 30, 38, 54, 70, 118, 126, 134, 150, 166, 182, 198, 214, 246, 278, 374, 534, 598, 662, 790, 854, 982, 1110, 1238, 1366, 1494, 1622, 1878, 2006, 2134, 2390, 2902, 3158, 3670, 5462, 5974, 6486, 6998, 10070, 11094, 12118

Offset: 1

Hugo Pfoertner and Rainer Rosenthal, Jan 06 2012

nonn

Positions of decreasing minima of abs(A203810(i)/A203811(i)).

			The first 4 fractions f(i)=A203810(i)/A203811(i) 1/1, 1/2, 1/6, -1/12 have decreasing absolute values. Therefore a(1)=1, a(2)=2, a(3)=3, a(4)=4. 5 is not in the sequence, because f(5)=7/60>1/12, but f(6)=-1/20 gives a(5)=6 because 1/20<1/12.
Fractions producing further decreasing absolute values are f(8)=-9/280, f(10)=-53/2520, f(12)=-373/27720, f(14)=-2869/360360, f(16)=-547/144144, f(18)=-1291/2450448, f(22)=-13913/232792560, f(30)=93259013/232908956280.

Hugo Pfoertner, Table of n, a(n) for n = 1..131

Cf. A203810, A203811.

s=0; d=2;\
for (k=1,12500,if(s>0,s-=1/k,s+=1/k);if(abs(s)Hugo Pfoertner, Nov 14 2017

A231693 Define a sequence of rationals by f(0)=0, thereafter f(n)=f(n-1)-1/n if that is >= 0, otherwise f(n)=f(n-1)+1/n; a(n) = denominator of f(n).

Original entry on oeis.org

1, 1, 2, 6, 12, 60, 20, 140, 280, 2520, 2520, 27720, 27720, 360360, 360360, 360360, 720720, 12252240, 4084080, 77597520, 77597520, 11085360, 11085360, 254963280, 84987760, 424938800, 424938800, 11473347600, 80313433200, 2329089562800, 2329089562800, 72201776446800, 144403552893600, 144403552893600

Offset: 0

N. J. A. Sloane, Nov 15 2013

See Comments in A231692, which is the sequence of numerators of {f(n)}.
Note that this sequence is not monotonic.
Differs from A002805 starting at a(20)=77597520: A002805(20)=15519504. See also A203811 for a very similar idea. - M. F. Hasler, Nov 15 2013

			0, 1, 1/2, 1/6, 5/12, 13/60, 1/20, 27/140, 19/280, 451/2520, 199/2520, 4709/27720, ...

David Wilson, Posting to Sequence Fans Mailing List, Nov 14 2013.

David W. Wilson, Table of n, a(n) for n = 0..200

Cf. A231692, A005132, A002805, A203811.

a231693 n = a231693_list !! n
a231693_list = map denominator $ 0 : wilson 1 0 where
   wilson x y = y' : wilson (x + 1) y'
                where y' = y + (if y < 1 % x then 1 else -1) % x
-- Reinhard Zumkeller, Nov 16 2013

f:=proc(n) option remember;
if n=0 then 0 elif
f(n-1) >= 1/n then f(n-1)-1/n else f(n-1)+1/n; fi; end;

Denominator[FoldList[# + (-1)^Boole[#*#2 >= 1]/#2 &, Range[0, 35]]] (* Paolo Xausa, Aug 16 2025 *)

s=0;vector(30,n,denominator(s-=(-1)^(n*s<1)/n)) \\ M. F. Hasler, Nov 15 2013

from fractions import Fraction
A231693 = [(f:=Fraction(0)).denominator] + [(f:=(f + (Fraction(1,i) if Fraction(1,i)>f else -Fraction(1,i)))).denominator for i in range(1, 34)]  # Jwalin Bhatt, Apr 08 2025

A232090 Minimal possible denominator for a sum of the form 1 +/- 1/2 +/- 1/3 +/- ... +/- 1/n.

Original entry on oeis.org

1, 2, 6, 12, 60, 20, 140, 280, 2520, 2520, 27720, 27720, 360360, 360360, 72072, 144144, 2450448, 272272, 5173168, 5173168, 739024, 739024, 16997552, 16997552, 424938800, 424938800, 11473347600, 11473347600, 332727080400, 332727080400

Offset: 1

M. F. Hasler, Nov 18 2013

nonn

Differs from A203811 from a(18)=272272 on, and from A002805 and A231693 from a(15)=72072 on.

Giovanni Resta, Table of n, a(n) for n = 1..37
F. T. Adams-Watters, Re: Reciprocal Recaman, SeqFan list, Nov 17 2013

Cf. A061195 (minimal possible positive numerator).

nMax = 19; d = {0}; Table[d = Flatten[{d + 1/n, d - 1/n}]; Min[Denominator[d]], {n, nMax}] (* T. D. Noe, Nov 19 2013 *)

for(n=0,19,m=(n+1)!;for(k=0,2^n-1,m=min(denominator(sum(j=2,n+1,(-1)^bittest(k,j-2)/j,1)),m));print1(m","))

Terms a(21)-a(30) from David W. Wilson, Nov 19 2013

cp's OEIS Frontend

A002805 Denominators of harmonic numbers H(n) = Sum_{i=1..n} 1/i.

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A203810 Numerators of s(i) = s(i-1) - (1/i)*sign(s(i-1)) with s(1) = 1.

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A203812 Numbers n where abs(s(n)) produces a new minimum, with s(1) = 1 and s(i) = s(i-1) - sign(s(i-1))*(1/i).

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A231693 Define a sequence of rationals by f(0)=0, thereafter f(n)=f(n-1)-1/n if that is >= 0, otherwise f(n)=f(n-1)+1/n; a(n) = denominator of f(n).

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A232090 Minimal possible denominator for a sum of the form 1 +/- 1/2 +/- 1/3 +/- ... +/- 1/n.

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