A000254 -id:A000254 - cp's OEIS frontend

A010051 Characteristic function of primes: 1 if n is prime, else 0.

0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0

Offset: 1

N. J. A. Sloane

The following sequences all have the same parity (with an extra zero term at the start of a(n)): a(n), A061007, A035026, A069754, A071574. - Jeremy Gardiner, Aug 09 2002
Hardy and Wright prove that the real number 0.011010100010... is irrational. See Nasehpour link. - Michel Marcus, Jun 21 2018
The spectral components (excluding the zero frequency) of the Fourier transform of the partial sequences {a(j)} with j=1..n and n an even number, exhibit a remarkable symmetry with respect to the central frequency component at position 1 + n/4. See the Fourier spectrum of the first 2^20 terms in Links, Comments in A289777, and Conjectures in A001223 of Sep 01 2019. It also appears that the symmetry grows with n. - Andres Cicuttin, Aug 23 2020

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 3.
V. Brun, Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare, Arch. Mat. Natur. B, 34, no. 8, 1915.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, London, 1975.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 65.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 132.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
Andres Cicuttin, Fourier spectrum of the first 2^20 terms of the characteristic function of primes.
William Craig, Jan-Willem van Ittersum and Ken Ono, Integer partitions detect the primes, PNAS, Vol. 121, No. 39 (2024), e2409417121.
Yoichi Motohashi, An overview of the Sieve Method and its History, arXiv:math/0505521 [math.NT], 2005-2006.
Peyman Nasehpour, A Computational Criterion for the Irrationality of Some Real Numbers, Journal of Algorithms and Computation, Vol. 52, No. 1 (2020), pp. 97-104, preprint, arXiv:1806.07560 [math.AC], 2018.
José L. Ramírez and Gustavo N. Rubiano, Properties and Generalizations of the Fibonacci Word Fractal, The Mathematica Journal, Vol. 16 (2014).
Eric Weisstein's World of Mathematics, Prime Number.
Eric Weisstein's World of Mathematics, Prime Constant.
Eric Weisstein's World of Mathematics, Prime zeta function primezeta(s).
Index entries for characteristic functions

Cf. A051006 (constant 0.4146825... (base 10) = 0.01101010001010001010... (base 2)), A001221 (inverse Moebius transform), A143519, A156660, A156659, A156657, A059500, A053176, A059456, A072762.
First differences of A000720, so A000720 gives partial sums.
Column k=1 of A117278.
Characteristic function of A000040.
Cf. A008683.

import Data.List (unfoldr)
a010051 :: Integer -> Int
a010051 n = a010051_list !! (fromInteger n-1)
a010051_list = unfoldr ch (1, a000040_list) where
   ch (i, ps'@(p:ps)) = Just (fromEnum (i == p),
                              (i + 1, if i == p then ps else ps'))
-- Reinhard Zumkeller, Apr 17 2012, Sep 15 2011

s:=[]; for n in [1..100] do if IsPrime(n) then s:=Append(s,1); else s:=Append(s,0); end if; end for; s;

[IsPrime(n) select 1 else 0: n in [1..100]];  // Bruno Berselli, Mar 02 2011

A010051:= n -> if isprime(n) then 1 else 0 fi;

Table[ If[ PrimeQ[n], 1, 0], {n, 105}] (* Robert G. Wilson v, Jan 15 2005 *)
Table[Boole[PrimeQ[n]], {n, 105}] (* Alonso del Arte, Aug 09 2011 *)
Table[PrimePi[n] - PrimePi[n-1], {n,50}] (* G. C. Greubel, Jan 05 2017 *)

a(n)=isprime(n) \\ Charles R Greathouse IV, Apr 16 2011

from sympy import isprime
def A010051(n): return int(isprime(n)) # Chai Wah Wu, Jan 20 2022

a(n) = floor(cos(Pi*((n-1)! + 1)/n)^2) for n >= 2. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 07 2002
Let M(n) be the n X n matrix m(i, j) = 0 if n divides ij + 1, m(i, j) = 1 otherwise; then for n > 0 a(n) = -det(M(n)). - Benoit Cloitre, Jan 17 2003
n >= 2, a(n) = floor(phi(n)/(n - 1)) = floor(A000010(n)/(n - 1)). - Benoit Cloitre, Apr 11 2003
a(n) = Sum_{d|gcd(n, A034386(n))} mu(d). [Brun]
a(m*n) = a(m)*0^(n - 1) + a(n)*0^(m - 1). - Reinhard Zumkeller, Nov 25 2004
a(n) = 1 if n has no divisors other than 1 and n, and 0 otherwise. - Jon Perry, Jul 02 2005
Dirichlet generating function: Sum_{n >= 1} a(n)/n^s = primezeta(s), where primezeta is the prime zeta function. - Franklin T. Adams-Watters, Sep 11 2005
a(n) = (n-1)!^2 mod n. - Franz Vrabec, Jun 24 2006
a(n) = A047886(n, 1). - Reinhard Zumkeller, Apr 15 2008
Equals A051731 (the inverse Möbius transform) * A143519. - Gary W. Adamson, Aug 22 2008
a(n) = A051731((n + 1)! + 1, n) from Wilson's theorem: n is prime if and only if (n + 1)! is congruent to -1 mod n. - N-E. Fahssi, Jan 20 2009, Jan 29 2009
a(n) = A166260/A001477. - Mats Granvik, Oct 10 2009
a(n) = 0^A070824, where 0^0=1. - Mats Granvik, Gary W. Adamson, Feb 21 2010
It appears that a(n) = (H(n)*H(n + 1)) mod n, where H(n) = n!*Sum_{k=1..n} 1/k = A000254(n). - Gary Detlefs, Sep 12 2010
Dirichlet generating function: log( Sum_{n >= 1} 1/(A112624(n)*n^s) ). - Mats Granvik, Apr 13 2011
a(n) = A100995(n) - sqrt(A100995(n)*A193056(n)). - Mats Granvik, Jul 15 2011
a(n) * (2 - n mod 4) = A151763(n). - Reinhard Zumkeller, Oct 06 2011
(n - 1)*a(n) = ( (2*n + 1)!! * Sum_{k=1..n}(1/(2*k + 1))) mod n, n > 2. - Gary Detlefs, Oct 07 2011
For n > 1, a(n) = floor(1/A001222(n)). - Enrique Pérez Herrero, Feb 23 2012
a(n) = mu(n) * Sum_{d|n} mu(d)*omega(d), where mu is A008683 and omega A001222 or A001221 indistinctly. - Enrique Pérez Herrero, Jun 06 2012
a(n) = A003418(n+1)/A003418(n) - A217863(n+1)/A217863(n) = A014963(n) - A072211(n). - Eric Desbiaux, Nov 25 2012
For n > 1, a(n) = floor(A014963(n)/n). - Eric Desbiaux, Jan 08 2013
a(n) = ((abs(n-2))! mod n) mod 2. - Timothy Hopper, May 25 2015
a(n) = abs(F(n)) - abs(F(n)-1/2) - abs(F(n)-1) + abs(f(n)-3/2), where F(n) = Sum_{m=2..n+1} (abs(1 - (n mod m)) - abs(1/2 - (n mod m)) + 1/2), n > 0. F(n) = 1 if n is prime, > 1 otherwise, except F(1) = 0. a(n) = 1 if F(n) = 1, 0 otherwise. - Timothy Hopper, Jun 16 2015
For n > 4, a(n) = (n-2)! mod n. - Thomas Ordowski, Jul 24 2016
From Ilya Gutkovskiy, Jul 24 2016: (Start)
G.f.: A(x) = Sum_{n>=1} x^A000040(n) = B(x)*(1 - x), where B(x) is the g.f. for A000720.
a(n) = floor(2/A000005(n)), for n>1. (End)
a(n) = pi(n) - pi(n-1) = A000720(n) - A000720(n-1), for n>=1. - G. C. Greubel, Jan 05 2017
Decimal expansion of Sum_{k>=1} (1/10)^prime(k) = 9 * Sum_{k>=1} pi(k)/10^(k+1), where pi(k) = A000720(k). - Amiram Eldar, Aug 11 2020
a(n) = 1 - ceiling((2/n) * Sum_{k=2..floor(sqrt(n))} floor(n/k)-floor((n-1)/k)), n>1. - Gary Detlefs, Sep 08 2023
a(n) = Sum_{d|n} mu(d)*omega(n/d), where mu = A008683 and omega = A001221. - Ridouane Oudra, Apr 12 2025
a(n) = 0 if (n^2 - 3*n + 2) * A000203(n) - 8 * A002127(n) > 0 else 1 (n>2, see Craig link). - Bill McEachen, Jul 04 2025

A001008 a(n) = numerator of harmonic number H(n) = Sum_{i=1..n} 1/i.

Original entry on oeis.org

1, 3, 11, 25, 137, 49, 363, 761, 7129, 7381, 83711, 86021, 1145993, 1171733, 1195757, 2436559, 42142223, 14274301, 275295799, 55835135, 18858053, 19093197, 444316699, 1347822955, 34052522467, 34395742267, 312536252003, 315404588903, 9227046511387

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.
By Wolstenholme's theorem, p^2 divides a(p-1) for all primes p > 3.
From Alexander Adamchuk, Dec 11 2006: (Start)
p divides a(p^2-1) for all primes p > 3.
p divides a((p-1)/2) for primes p in A001220.
p divides a((p+1)/2) or a((p-3)/2) for primes p in A125854.
a(n) is prime for n in A056903. Corresponding primes are given by A067657. (End)
a(n+1) is the numerator of the polynomial A[1, n](1) where the polynomial A[genus 1, level n](m) is defined to be Sum_{d = 1..n - 1} m^(n - d)/d. (See the Mathematica procedure generating A[1, n](m) below.) - Artur Jasinski, Oct 16 2008
Better solutions to the card stacking problem have been found by M. Paterson and U. Zwick (see link). - Hugo Pfoertner, Jan 01 2012
a(n) = A213999(n, n-1). - Reinhard Zumkeller, Jul 03 2012
a(n) coincides with A175441(n) if and only if n is not from the sequence A256102. The quotient a(n) / A175441(n) for n in A256102 is given as corresponding entry of A256103. - Wolfdieter Lang, Apr 23 2015
For a very short proof that the Harmonic series diverges, see the Goldmakher link. - N. J. A. Sloane, Nov 09 2015
All terms are odd while corresponding denominators (A002805) are all even for n > 1 (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, ... ].
Coincidences with A175441: the first 19 entries coincide because 20 is the first entry of A256102. Indeed, a(20)/A175441(20) = 55835135 / 11167027 = 5 = A256103(1). - _Wolfdieter Lang_, Apr 23 2015

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 258-261.
H. W. Gould, Combinatorial Identities, Morgantown Printing and Binding Co., 1972, # 1.45, page 6, #3.122, page 36.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 259.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 347.
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.
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..2295 (first 200 terms provided by T. D. Noe)
David H. Bailey, Jonathan M. Borwein, and Roland Girgensohn, Experimental evaluation of Euler sums, Exper. Math. 3(1) (1994), 17-30; they evaluate the constants Sum_{k>=1} H_k^m/(k+1)^n.
Hongwei Chen, Evaluations of Some Variant Euler Sums, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.3.
Hongwei Chen, Interesting Series Associated with Central Binomial Coefficients, Catalan Numbers and Harmonic Numbers, J. Int. Seq. 19 (2016), #16.1.5.
R. M. Dickau, Harmonic numbers and the book-stacking problem.
Leo Goldmakher, A short(er) proof of the divergence of the harmonic series.
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.
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
Romeo Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv preprint arXiv:1111.3057 [math.NT], 2011.
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
Hisanori Mishima, Factorizations of Wolstenholme numbers, n=1..100, n=101..200, n=201..300.
Mike Paterson et al., Maximum Overhang.
Maxie D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications, J. Int. Seq. 13 (2010), 10.6.7, Section 4.3.
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.
Roberto Tauraso, Some Congruences for Central Binomial Sums Involving Fibonacci and Lucas Numbers, JIS 19 (2016) # 16.5.4.
Eric Weisstein's World of Mathematics, Book Stacking Problem, Wolstenholme's Theorem, Harmonic Mean, Digamma Function.
Wikipedia, Harmonic number.

Cf. A002805 (denominators), A007406, A007408, A007410, A075135, A001220, A125854, A121999, A014566, A056903, A067657, A177427, A177690.
Cf. A145609-A145640. - Artur Jasinski, Oct 16 2008
Cf. A003506. - Paul Curtz, Nov 30 2013
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.
Cf. A195505.

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

import Data.Ratio ((%), numerator)
a001008 = numerator . sum . map (1 %) . enumFromTo 1
a001008_list = map numerator $ scanl1 (+) $ map (1 %) [1..]
-- Reinhard Zumkeller, Jul 03 2012

[Numerator(HarmonicNumber(n)): n in [1..30]]; // Bruno Berselli, Feb 17 2016

A001008 := proc(n)
    add(1/k,k=1..n) ;
    numer(%) ;
end proc:
seq( A001008(n),n=1..40) ; # Zerinvary Lajos, Mar 28 2007; R. J. Mathar, Dec 02 2016

Table[Numerator[HarmonicNumber[n]], {n, 30}]
(* Procedure generating A[1,n](m) (see Comments section) *) m =1; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, k], {r, 1, 20}]; aa (* Artur Jasinski, Oct 16 2008 *)
Numerator[Accumulate[1/Range[25]]] (* Alonso del Arte, Nov 21 2018 *)
Numerator[Table[((n - 1)/2)*HypergeometricPFQ[{1, 1, 2 - n}, {2, 3}, 1] + 1, {n, 1, 29}]] (* Artur Jasinski, Jan 08 2021 *)

A001008(n) = numerator(sum(i=1,n,1/i)) \\ Michael B. Porter, Dec 08 2009

H1008=List(1); A001008(n)={for(k=#H1008,n-1,listput(H1008,H1008[k]+1/(k+1))); numerator(H1008[n])} \\ about 100x faster for n=1..1500. - M. F. Hasler, Jul 03 2019

from sympy import Integer
[sum(1/Integer(i) for i in range(1, n + 1)).numerator() for n in range(1, 31)]  # Indranil Ghosh, Mar 23 2017

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 A001008(n): H = harmonic(1,n+1); return numerator(H[0]/H[1])
[A001008(n) for n in (1..29)] # Peter Luschny, Sep 01 2012

H(n) ~ log n + gamma + O(1/n). [See Hardy and Wright, Th. 422.]
log n + gamma - 1/n < H(n) < log n + gamma + 1/n [follows easily from Hardy and Wright, Th. 422]. - David Applegate and N. J. A. Sloane, Oct 14 2008
G.f. for H(n): log(1-x)/(x-1). - Benoit Cloitre, Jun 15 2003
H(n) = sqrt(Sum_{i = 1..n} Sum_{j = 1..n} 1/(i*j)). - Alexander Adamchuk, Oct 24 2004
a(n) is the numerator of Gamma/n + Psi(1 + n)/n = Gamma + Psi(n), where Psi is the digamma function. - Artur Jasinski, Nov 02 2008
H(n) = 3/2 + 2*Sum_{k = 0..n-3} binomial(k+2, 2)/((n-2-k)*(n-1)*n), n > 1. - Gary Detlefs, Aug 02 2011
H(n) = (-1)^(n-1)*(n+1)*n*Sum_{k = 0..n-1} k!*Stirling2(n-1, k) * Stirling1(n+k+1,n+1)/(n+k+1)!. - Vladimir Kruchinin, Feb 05 2013
H(n) = n*Sum_{k = 0..n-1} (-1)^k*binomial(n-1,k)/(k+1)^2. (Wenchang Chu) - Gary Detlefs, Apr 13 2013
H(n) = (1/2)*Sum_{k = 1..n} (-1)^(k-1)*binomial(n,k)*binomial(n+k, k)/k. (H. W. Gould) - Gary Detlefs, Apr 13 2013
E.g.f. for H(n) = a(n)/A002805(n): (gamma + log(x) - Ei(-x)) * exp(x), where gamma is the Euler-Mascheroni constant, and Ei(x) is the exponential integral. - Vladimir Reshetnikov, Apr 24 2013
H(n) = residue((psi(-s)+gamma)^2/2, {s, n}) where psi is the digamma function and gamma is the Euler-Mascheroni constant. - Jean-François Alcover, Feb 19 2014
H(n) = Sum_{m >= 1} n/(m^2 + n*m) = gamma + digamma(1+n), numerators and denominators. (see Mathworld link on Digamma). - Richard R. Forberg, Jan 18 2015
H(n) = (1/2) Sum_{j >= 1} Sum_{k = 1..n} ((1 - 2*k + 2*n)/((-1 + k + j*n)*(k + j*n))) + log(n) + 1/(2*n). - Dimitri Papadopoulos, Jan 13 2016
H(n) = (n!)^2*Sum_{k = 1..n} 1/(k*(n-k)!*(n+k)!). - Vladimir Kruchinin, Mar 31 2016
a(n) = Stirling1(n+1, 2) / gcd(Stirling1(n+1, 2), n!) = A000254(n) / gcd(A000254(n), n!). - Max Alekseyev, Mar 01 2018
From Peter Bala, Jan 31 2019: (Start)
H(n) = 1 + (1 + 1/2)*(n-1)/(n+1) + (1/2 + 1/3)*(n-1)*(n-2)/((n+1)*(n+2)) + (1/3 + 1/4)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + ... .
H(n)/n  = 1 + (1/2^2 - 1)*(n-1)/(n+1) + (1/3^2 - 1/2^2)*(n-1)*(n-2)/((n+1)*(n+2)) + (1/4^2 - 1/3^2)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + ... .
For odd n >= 3, (1/2)*H((n-1)/2) = (n-1)/(n+1) + (1/2)*(n-1)*(n-3)/((n+1)*(n+3)) + 1/3*(n-1)*(n-3)*(n-5)/((n+1)*(n+3)*(n+5)) + ... . Cf. A195505. See the Bala link in A036970. (End)
H(n) = ((n-1)/2) * hypergeom([1,1,2-n], [2,3], 1) + 1. - Artur Jasinski, Jan 08 2021
Conjecture: for nonzero m, H(n) = (1/m)*Sum_{k = 1..n} ((-1)^(k+1)/k) * binomial(m*k,k)*binomial(n+(m-1)*k,n-k). The case m = 1 is well-known; the case m = 2 is given above by Detlefs (dated Apr 13 2013). - Peter Bala, Mar 04 2022
a(n) = the (reduced) numerator 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
H(n) = Sum_{k=1..n} (-1)^(k-1)*binomial(n,k)/k (H. W. Gould). - Gary Detlefs, May 28 2024

Edited by Max Alekseyev, Oct 21 2011

Changed title, deleting the incorrect name "Wolstenholme numbers" which conflicted with the definition of the latter in both Weisstein's World of Mathematics and in Wikipedia, as well as with OEIS A007406. - Stanislav Sykora, Mar 25 2016

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

Original entry on oeis.org

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

A000272 Number of trees on n labeled nodes: n^(n-2) with a(0)=1.

Original entry on oeis.org

1, 1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000, 2357947691, 61917364224, 1792160394037, 56693912375296, 1946195068359375, 72057594037927936, 2862423051509815793, 121439531096594251776, 5480386857784802185939, 262144000000000000000000, 13248496640331026125580781

Offset: 0

N. J. A. Sloane

Number of spanning trees in complete graph K_n on n labeled nodes.
Robert Castelo, Jan 06 2001, observes that n^(n-2) is also the number of transitive subtree acyclic digraphs on n-1 vertices.
a(n) is also the number of ways of expressing an n-cycle in the symmetric group S_n as a product of n-1 transpositions, see example. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001
Also counts parking functions, critical configurations of the chip firing game, allowable pairs sorted by a priority queue [Hamel].
The parking functions of length n can be described as all permutations of all words [d(1),d(2), ..., d(n)] where 1 <= d(k) <= k; see example. There are (n+1)^(n-1) = a(n+1) parking functions of length n. - Joerg Arndt, Jul 15 2014
a(n+1) is the number of endofunctions with no cycles of length > 1; number of forests of rooted labeled trees on n vertices. - Mitch Harris, Jul 06 2006
a(n) is also the number of nilpotent partial bijections (of an n-element set). Equivalently, the number of nilpotents in the partial symmetric semigroup, P sub n. - Abdullahi Umar, Aug 25 2008
a(n) is also the number of edge-labeled rooted trees on n nodes. - Nikos Apostolakis, Nov 30 2008
a(n+1) is the number of length n sequences on an alphabet of {1,2,...,n} that have a partial sum equal to n. For example a(4)=16 because there are 16 length 3 sequences on {1,2,3} in which the terms (beginning with the first term and proceeding sequentially) sum to 3 at some point in the sequence. {1, 1, 1}, {1, 2, 1}, {1, 2, 2}, {1, 2, 3}, {2, 1, 1}, {2, 1, 2}, {2, 1, 3}, {3, 1, 1}, {3, 1, 2}, {3, 1, 3}, {3, 2, 1}, {3, 2, 2}, {3, 2, 3}, {3, 3, 1}, {3, 3, 2}, {3, 3, 3}. - Geoffrey Critzer, Jul 20 2009
a(n) is the number of acyclic functions from {1,2,...,n-1} to {1,2,...,n}. An acyclic function f satisfies the following property: for any x in the domain, there exists a positive integer k such that (f^k)(x) is not in the domain. Note that f^k denotes the k-fold composition of f with itself, e.g., (f^2)(x)=f(f(x)). - Dennis P. Walsh, Mar 02 2011
a(n) is the absolute value of the discriminant of the polynomial x^{n-1}+...+x+1. More precisely, a(n) = (-1)^{(n-1)(n-2)/2} times the discriminant. - Zach Teitler, Jan 28 2014
For n > 2, a(n+2) is the number of nodes in the canonical automaton for the affine Weyl group of type A_n. - Tom Edgar, May 12 2016
The tree formula a(n) = n^(n-2) is due to Cayley (see the first comment). - Jonathan Sondow, Jan 11 2018
a(n) is the number of topologically distinct lines of play for the game Planted Brussels Sprouts on n vertices. See Ji and Propp link. - Caleb Ji, May 11 2018
a(n+1) is also the number of bases of R^n, that can be made from the n(n+1)/2 vectors of the form [0 ... 0 1 ... 1 0 ... 0]^T, where the initial or final zeros are optional, but at least one 1 has to be included. - Nicolas Nagel, Jul 31 2018
Cooper et al. show that every connected k-chromatic graph contains at least k^(k-2) spanning trees. - Michel Marcus, May 14 2020

			a(7)=matdet([196, 175, 140, 98, 56, 21; 175, 160, 130, 92, 53, 20; 140, 130, 110, 80, 47, 18; 98, 92, 80, 62, 38, 15; 56, 53, 47, 38, 26, 11; 21, 20, 18, 15, 11, 6])=16807
a(3)=3 since there are 3 acyclic functions f:[2]->[3], namely, {(1,2),(2,3)}, {(1,3),(2,1)}, and {(1,3),(2,3)}.
From _Joerg Arndt_ and Greg Stevenson, Jul 11 2011: (Start)
The following products of 3 transpositions lead to a 4-cycle in S_4:
  (1,2)*(1,3)*(1,4);
  (1,2)*(1,4)*(3,4);
  (1,2)*(3,4)*(1,3);
  (1,3)*(1,4)*(2,3);
  (1,3)*(2,3)*(1,4);
  (1,4)*(2,3)*(2,4);
  (1,4)*(2,4)*(3,4);
  (1,4)*(3,4)*(2,3);
  (2,3)*(1,2)*(1,4);
  (2,3)*(1,4)*(2,4);
  (2,3)*(2,4)*(1,2);
  (2,4)*(1,2)*(3,4);
  (2,4)*(3,4)*(1,2);
  (3,4)*(1,2)*(1,3);
  (3,4)*(1,3)*(2,3);
  (3,4)*(2,3)*(1,2).  (End)
The 16 parking functions of length 3 are 111, 112, 121, 211, 113, 131, 311, 221, 212, 122, 123, 132, 213, 231, 312, 321. - _Joerg Arndt_, Jul 15 2014
G.f. = 1 + x + x^2 + 3*x^3 + 16*x^4 + 125*x^5 + 1296*x^6 + 16807*x^7 + ...

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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 25, Prop. 5.3.2.
J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge Univ. Press, 1992.

Seiichi Manyama, Table of n, a(n) for n = 0..388 (terms 0..100 from N. J. A. Sloane)
Federico Ardila, Matthias Beck, and Jodi McWhirter, The Arithmetic of Coxeter Permutahedra, arXiv:2004.02952 [math.CO], 2020.
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Norman L. Biggs, Chip-firing and the critical group of a graph, J. Algeb. Combin., 9 (1999), 25-45.
Norman L. Biggs, E. Keith Lloyd, and Robin J. Wilson, Graph Theory 1736-1936, Oxford, 1976, p. 51.
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Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
David Callan, A Combinatorial Derivation of the Number of Labeled Forests, J. Integer Seqs., Vol. 6, 2003.
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Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions. [Broken link]
R. Castelo and A. Siebes, A characterization of moral transitive directed acyclic graph Markov models as labeled trees, Report CS-2000-44, Department of Computer Science, Univ. Utrecht.
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Jacob W. Cooper, Adam Kabela, Daniel Král', and Théo Pierron, Hadwiger meets Cayley, arXiv:2005.05989 [math.CO], 2020.
S. Coulomb and M. Bauer, On vertex covers, matchings and random trees, arXiv:math/0407456 [math.CO], 2004.
FindStat - Combinatorial Statistic Finder, Parking functions
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Index entries for sequences related to trees
Index entries for "core" sequences

Cf. A000169, A000254, A000312, A007778, A007830, A008785-A008791, A052750, A081048, A083483, A097170, A239910.
a(n) = A033842(n-1, 0) (first column of triangle).
a(n) = A058127(n-1, n) (right edge of triangle).
Cf. A000272 (labeled trees), A036361 (labeled 2-trees), A036362 (labeled 3-trees), A036506 (labeled 4-trees), A000055 (unlabeled trees), A054581 (unlabeled 2-trees).
Column m=1 of A105599. - Alois P. Heinz, Apr 10 2014

a000272 0 = 1; a000272 1 = 1
a000272 n = n ^ (n - 2)  -- Reinhard Zumkeller, Jul 07 2013

[ n^(n-2) : n in [1..10]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

A000272 := n -> ifelse(n=0, 1, n^(n-2)): seq(A000272(n), n = 0..20); # Peter Luschny, Jun 12 2022

<< DiscreteMath`Combinatorica` Table[NumberOfSpanningTrees[CompleteGraph[n]], {n, 1, 20}] (* Artur Jasinski, Dec 06 2007 *)
Join[{1},Table[n^(n-2),{n,20}]] (* Harvey P. Dale, Nov 28 2012 *)
a[ n_] := If[ n < 1, Boole[n == 0], n^(n - 2)]; (* Michael Somos, May 25 2014 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 - LambertW[-x] - LambertW[-x]^2 / 2, {x, 0, n}]]; (* Michael Somos, May 25 2014 *)
a[ n_] := If[ n < 1, Boole[n == 0], With[ {m = n - 1}, m! SeriesCoefficient[ Exp[ -LambertW[-x]], {x, 0, m}]]]; (* Michael Somos, May 25 2014 *)
a[ n_] := If[ n < 2, Boole[n >= 0], With[ {m = n - 1}, m! SeriesCoefficient[ InverseSeries[ Series[ Log[1 + x] / (1 + x), {x, 0, m}]], m]]]; (* Michael Somos, May 25 2014 *)
a[ n_] := If[ n < 1, Boole[n == 0], With[ {m = n - 1}, m! SeriesCoefficient[ Nest[ 1 + Integrate[ #^2 / (1 - x #), x] &, 1 + O[x], m], {x, 0, m}]]]; (* Michael Somos, May 25 2014 *)

A000272[n]:=if n=0 then 1 else n^(n-2)$
makelist(A000272[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

{a(n) = if( n<1, n==0, n^(n-2))}; /* Michael Somos, Feb 16 2002 */

{a(n) = my(A); if( n<1, n==0, n--; A = 1 + O(x); for(k=1, n, A = 1 + intformal( A^2 / (1 - x * A))); n! * polcoeff( A, n))}; /* Michael Somos, May 25 2014 */

/* GP Function for Determinant of Hermitian (square symmetric) matrix for univariate polynomial of degree n by Gerry Martens: */
Hn(n=2)= {local(H=matrix(n-1,n-1),i,j); for(i=1,n-1, for(j=1,i, H[i,j]=(n*i^3-3*n*(n+1)*i^2/2+n*(3*n+1)*i/2+(n^4-n^2)/2)/6-(i^2-(2*n+1)*i+n*(n+1))*(j-1)*j/4; H[j,i]=H[i,j]; ); ); print("a(",n,")=matdet(",H,")"); print("Determinant H =",matdet(H)); return(matdet(H)); } { print(Hn(7)); } /* Gerry Martens, May 04 2007 */

def A000272(n): return 1 if n <= 1 else n**(n-2) # Chai Wah Wu, Feb 03 2022

E.g.f.: 1 + T - (1/2)*T^2; where T=T(x) is Euler's tree function (see A000169, also A001858). - Len Smiley, Nov 19 2001
Number of labeled k-trees on n nodes is binomial(n, k) * (k*(n-k)+1)^(n-k-2).
E.g.f. for b(n)=a(n+2): ((W(-x)/x)^2)/(1+W(-x)), where W is Lambert's function (principal branch). [Equals d/dx (W(-x)/(-x)). - Wolfdieter Lang, Oct 25 2022]
Determinant of the symmetric matrix H generated for a polynomial of degree n by: for(i=1,n-1, for(j=1,i, H[i,j]=(n*i^3-3*n*(n+1)*i^2/2+n*(3*n+1)*i/2+(n^4-n^2)/2)/6-(i^2-(2*n+1)*i+n*(n+1))*(j-1)*j/4; H[j,i]=H[i,j]; ); );. - Gerry Martens, May 04 2007
a(n+1) = Sum_{i=1..n} i * n^(n-1-i) * binomial(n, i). - Yong Kong (ykong(AT)curagen.com), Dec 28 2000
For n >= 1, a(n+1) = Sum_{i=1..n} n^(n-i)*binomial(n-1,i-1). - Geoffrey Critzer, Jul 20 2009
E.g.f. for b(n)=a(n+1): exp(-W(-x)), where W is Lambert's function satisfying W(x)*exp(W(x))=x. Proof is contained in link "Notes on acyclic functions..." - Dennis P. Walsh, Mar 02 2011
From Sergei N. Gladkovskii, Sep 18 2012: (Start)
E.g.f.: 1 + x + x^2/(U(0) - x) where U(k) = x*(k+1)*(k+2)^k + (k+1)^k*(k+2) - x*(k+2)^2*(k+3)*((k+1)*(k+3))^k/U(k+1); (continued fraction).
G.f.: 1 + x + x^2/(U(0)-x) where U(k) = x*(k+1)*(k+2)^k + (k+1)^k - x*(k+2)*(k+3)*((k+1)*(k+3))^k/E(k+1); (continued fraction). (End)
Related to A000254 by Sum_{n >= 1} a(n+1)*x^n/n! = series reversion( 1/(1 + x)*log(1 + x) ) = series reversion(x - 3*x^2/2! + 11*x^3/3! - 50*x^4/4! + ...). Cf. A052750. - Peter Bala, Jun 15 2016
For n >= 3 and 2 <= k <= n-1, the number of trees on n vertices with exactly k leaves is binomial(n,k)*S(n-2,n-k)(n-k)! where S(a,b) is the Stirling number of the second kind. Therefore a(n) = Sum_{k=2..n-1} binomial(n,k)*S(n-2,n-k)(n-k)! for n >= 3. - Jonathan Noel, May 05 2017

A001710 Order of alternating group A_n, or number of even permutations of n letters.

Original entry on oeis.org

1, 1, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000, 1216451004088320000, 25545471085854720000, 562000363888803840000

Offset: 0

N. J. A. Sloane

For n >= 3, a(n-1) is also the number of ways that a 3-cycle in the symmetric group S_n can be written as a product of 2 long cycles (of length n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 14 2001
a(n) is the number of Hamiltonian circuit masks for an n X n adjacency matrix of an undirected graph. - Chad Brewbaker, Jan 31 2003
a(n-1) is the number of necklaces one can make with n distinct beads: n! bead permutations, divide by two to represent flipping the necklace over, divide by n to represent rotating the necklace. Related to Stirling numbers of the first kind, Stirling cycles. - Chad Brewbaker, Jan 31 2003
Number of increasing runs in all permutations of [n-1] (n>=2). Example: a(4)=12 because we have 12 increasing runs in all the permutations of [3] (shown in parentheses): (123), (13)(2), (3)(12), (2)(13), (23)(1), (3)(2)(1). - Emeric Deutsch, Aug 28 2004
Minimum permanent over all n X n (0,1)-matrices with exactly n/2 zeros. - Simone Severini, Oct 15 2004
The number of permutations of 1..n that have 2 following 1 for n >= 1 is 0, 1, 3, 12, 60, 360, 2520, 20160, ... . - Jon Perry, Sep 20 2008
Starting (1, 3, 12, 60, ...) = binomial transform of A000153: (1, 2, 7, 32, 181, ...). - Gary W. Adamson, Dec 25 2008
First column of A092582. - Mats Granvik, Feb 08 2009
The asymptotic expansion of the higher order exponential integral E(x,m=1,n=3) ~ exp(-x)/x*(1 - 3/x + 12/x^2 - 60/x^3 + 360/x^4 - 2520/x^5 + 20160/x^6 - 81440/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009
For n>1: a(n) = A173333(n,2). - Reinhard Zumkeller, Feb 19 2010
Starting (1, 3, 12, 60, ...) = eigensequence of triangle A002260, (a triangle with k terms of (1,2,3,...) in each row given k=1,2,3,...). Example: a(6) = 360, generated from (1, 2, 3, 4, 5) dot (1, 1, 3, 12, 60) = (1 + 2 + 9 + 48 + 300). - Gary W. Adamson, Aug 02 2010
For n>=2: a(n) is the number of connected 2-regular labeled graphs on (n+1) nodes (Cf. A001205). - Geoffrey Critzer, Feb 16 2011.
The Fi1 and Fi2 triangle sums of A094638 are given by the terms of this sequence (n>=1). For the definition of these triangle sums see A180662. - Johannes W. Meijer, Apr 20 2011
Also [1, 1] together with the row sums of triangle A162608. - Omar E. Pol, Mar 09 2012
a(n-1) is, for n>=2, also the number of necklaces with n beads (only C_n symmetry, no turnover) with n-1 distinct colors and signature c[.]^2 c[.]^(n-2). This means that two beads have the same color, and for n=2 the second factor is omitted. Say, cyclic(c[1]c[1]c[2]c[3]..c[n-1]), in short 1123...(n-1), taken cyclically. E.g., n=2: 11,  n=3: 112, n=4: 1123, 1132, 1213,  n=5: 11234, 11243, 11324, 11342, 11423, 11432, 12134, 12143, 13124, 13142, 14123, 14132. See the next-to-last entry in line n>=2 of the representative necklace partition array A212359. - Wolfdieter Lang, Jun 26 2012
For m >= 3, a(m-1) is the number of distinct Hamiltonian circuits in a complete simple graph with m vertices. See also A001286. - Stanislav Sykora, May 10 2014
In factorial base (A007623) these numbers have a simple pattern: 1, 1, 1, 11, 200, 2200, 30000, 330000, 4000000, 44000000, 500000000, 5500000000, 60000000000, 660000000000, 7000000000000, 77000000000000, 800000000000000, 8800000000000000, 90000000000000000, 990000000000000000, etc. See also the formula based on this observation, given below. - Antti Karttunen, Dec 19 2015
Also (by definition) the independence number of the n-transposition graph. - Eric W. Weisstein, May 21 2017
Number of permutations of n letters containing an even number of even cycles. - Michael Somos, Jul 11 2018
Equivalent to Brewbaker's and Sykora's comments, a(n - 1) is the number of undirected cycles covering n labeled vertices, hence the logarithmic transform of A002135. - Gus Wiseman, Oct 20 2018
For n >= 2 and a set of n distinct leaf labels, a(n) is the number of binary, rooted, leaf-labeled tree topologies that have a caterpillar shape (column k=1 of A306364). - Noah A Rosenberg, Feb 11 2019
Also the clique covering number of the n-Bruhat graph. - Eric W. Weisstein, Apr 19 2019
a(n) is the number of lattices of the form [s,w] in the weak order on S_n, for a fixed simple reflection s. - Bridget Tenner, Jan 16 2020
For n > 3, a(n) = p_1^e_1*...*p_m^e_m, where p_1 = 2 and e_m = 1. There exists p_1^x where x <= e_1 such that p_1^x*p_m^e_m is a primitive Zumkeller number (A180332) and p_1^e_1*p_m^e_m is a Zumkeller number (A083207). Therefore, for n > 3, a(n) = p_1^e_1*p_m^e_m*r, where r is relatively prime to p_1*p_m, is also a Zumkeller number. - Ivan N. Ianakiev, Mar 11 2020
For n>1, a(n) is the number of permutations of [n] that have 1 and 2 as cycle-mates, that is, 1 and 2 are contained in the same cycle of a cyclic representation of permutations of [n]. For example, a(4) counts the 12 permutations with 1 and 2 as cycle-mates, namely, (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), (1 4 3 2), (1 2 3) (4), (1 3 2) (4), (1 2 4 )(3), (1 4 2)(3), (1 2)(3 4), and (1 2)(3)(4). Since a(n+2)=row sums of A162608, our result readily follows. - Dennis P. Walsh, May 28 2020

			G.f. = 1 + x + x^2 + 3*x^3 + 12*x^4 + 60*x^5 + 360*x^6 + 2520*x^7 + ...

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 87-8, 20. (a), c_n^e(t=1).
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).

N. J. A. Sloane, Table of n, a(n) for n = 0..100
Somaya Barati, Beáta Bényi, Abbas Jafarzadeh, and Daniel Yaqubi, Mixed restricted Stirling numbers, arXiv:1812.02955 [math.CO], 2018.
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Jonathan Beagley and Lara Pudwell, Colorful Tilings and Permutations, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
Olivier Bodini, Antoine Genitrini, and Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.
Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), Article 00.1.5.
Camille Combe and Samuele Giraudo, Cliff operads: a hierarchy of operads on words, arXiv:2106.14552 [math.CO], 2021.
Mareike Fischer, Extremal Values of the Sackin Tree Balance Index, Ann. Comb. (2021) Vol. 25, 515-541, Remark 7.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 36.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 262.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Shirali Kadyrov and Farukh Mashurov, Generalized continued fraction expansions for Pi and E, arXiv:1912.03214 [math.NT], 2019.
Chanchal Kumar and Amit Roy, Integer Sequences and Monomial Ideals, arXiv:2003.10098 [math.CO], 2020.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Xah Lee, Combinatorics: Loop in n points.
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
S-Z Song, S-G Hwang, S-H Rim, and G-S Cheon, Extremes of permanents of (0,1)-matrices, Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), 197-210.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
B. E. Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Alternating Group.
Eric Weisstein's World of Mathematics, Bruhat Graph.
Eric Weisstein's World of Mathematics, Circular Permutation.
Eric Weisstein's World of Mathematics, Clique Covering Number.
Eric Weisstein's World of Mathematics, Even Permutation.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Eric Weisstein's World of Mathematics, Independence Number.
Eric Weisstein's World of Mathematics, Odd Permutation.
Eric Weisstein's World of Mathematics, Transposition Graph.
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 7.
Index entries for sequences related to factorial base representation.
Index entries for sequences related to factorial numbers.
Index entries for sequences related to groups.
Index to divisibility sequences.

Cf. A000142, A000153, A000255, A001147, A001286, A001720, A002135, A002260, A007623, A007717, A049444, A049459, A093468, A094587, A094638, A138533, A153880, A173333, A213936, A215771, A319225, A319226, A320655.
a(n+1)= A046089(n, 1), n >= 1 (first column of triangle), A161739 (q(n) sequence).
Bisections are A002674 and A085990 (essentially).
Row 3 of A265609 (essentially).
Row sums of A307429.

[1] cat [Order(AlternatingGroup(n)): n in [1..20]]; // Arkadiusz Wesolowski, May 17 2014

seq(mul(k, k=3..n), n=0..20); # Zerinvary Lajos, Sep 14 2007

a[n_]:= If[n > 2, n!/2, 1]; Array[a, 21, 0]
a[n_]:= If[n<3, 1, n*a[n-1]]; Array[a, 21, 0]; (* Robert G. Wilson v, Apr 16 2011 *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[(2-x^2)/(2-2x), {x, 0, n}]]; (* Michael Somos, May 22 2014 *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[1 +Sinh[-Log[1-x]], {x, 0, n}]]; (* Michael Somos, May 22 2014 *)
Numerator[Range[0, 20]!/2] (* Eric W. Weisstein, May 21 2017 *)
Table[GroupOrder[AlternatingGroup[n]], {n, 0, 20}] (* Eric W. Weisstein, May 21 2017 *)

PARI
```
{a(n) = if( n<2, n>=0, n!/2)};
```

a(n)=polcoeff(1+x*sum(m=0,n,m^m*x^m/(1+m*x+x*O(x^n))^m),n) \\ Paul D. Hanna

A001710=n->n!\2+(n<2) \\ M. F. Hasler, Dec 01 2013

from math import factorial
def A001710(n): return factorial(n)>>1 if n > 1 else 1 # Chai Wah Wu, Feb 14 2023

def A001710(n): return (factorial(n) +int(n<2))//2
[A001710(n) for n in range(31)] # G. C. Greubel, Sep 28 2024

;; Using memoization-macro definec for which an implementation can be found in http://oeis.org/wiki/Memoization
(definec (A001710 n) (cond ((<= n 2) 1) (else (* n (A001710 (- n 1))))))
;; Antti Karttunen, Dec 19 2015

a(n) = numerator(n!/2) and A141044(n) = denominator(n!/2).
D-finite with recurrence: a(0) = a(1) = a(2) = 1; a(n) = n*a(n-1) for n>2. - Chad Brewbaker, Jan 31 2003 [Corrected by N. J. A. Sloane, Jul 25 2008]
a(0) = 0, a(1) = 1; a(n) = Sum_{k=1..n-1} k*a(k). - Amarnath Murthy, Oct 29 2002
Stirling transform of a(n+1) = [1, 3, 12, 160, ...] is A083410(n) = [1, 4, 22, 154, ...]. - Michael Somos, Mar 04 2004
First Eulerian transform of A000027. See A000142 for definition of FET. - Ross La Haye, Feb 14 2005
From Paul Barry, Apr 18 2005: (Start)
a(n) = 0^n + Sum_{k=0..n} (-1)^(n-k-1)*T(n-1, k)*cos(Pi*(n-k-1)/2)^2.
T(n,k) = abs(A008276(n, k)). (End)
E.g.f.: (2 - x^2)/(2 - 2*x).
E.g.f. of a(n+2), n>=0, is 1/(1-x)^3.
E.g.f.: 1 + sinh(log(1/(1-x))). - Geoffrey Critzer, Dec 12 2010
a(n+1) = (-1)^n * A136656(n,1), n>=1.
a(n) = n!/2 for n>=2 (proof from the e.g.f). - Wolfdieter Lang, Apr 30 2010
a(n) = (n-2)! * t(n-1), n>1, where t(n) is the n-th triangular number (A000217). - Gary Detlefs, May 21 2010
a(n) = ( A000254(n) - 2* A001711(n-3) )/3, n>2. - Gary Detlefs, May 24 2010
O.g.f.: 1 + x*Sum_{n>=0} n^n*x^n/(1 + n*x)^n. - Paul D. Hanna, Sep 13 2011
a(n) = if n < 2 then 1, otherwise Pochhammer(n,n)/binomial(2*n,n). - Peter Luschny, Nov 07 2011
a(n) = Sum_{k=0..floor(n/2)} s(n,n-2*k) where s(n,k) are Stirling number of the first kind, A048994. - Mircea Merca, Apr 07 2012
a(n-1), n>=3, is M_1([2,1^(n-2)])/n = (n-1)!/2, with the M_1 multinomial numbers for the given n-1 part partition of n. See the second to last entry in line n>=3 of A036038, and the above necklace comment by W. Lang. - Wolfdieter Lang, Jun 26 2012
G.f.: A(x) = 1 + x + x^2/(G(0)-2*x) where G(k) = 1 - (k+1)*x/(1 - x*(k+3)/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Dec 26 2012.
G.f.: 1 + x + (Q(0)-1)*x^2/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+2)*sqrt(x)/(1 - sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x + (x*Q(x)-x^2)/(2*(sqrt(x)+x)), where Q(x) = Sum_{n>=0} (n+1)!*x^n*sqrt(x)*(sqrt(x) + x*(n+2)). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x/2 + (Q(0)-1)*x/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+1)*sqrt(x)/(1 - sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x + x^2*G(0)/2, where G(k) = 1 + 1/(1 - x/(x + 1/(k+3)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
G.f.: 1+x + x^2*W(0), where W(k) = 1 - x*(k+3)/( x*(k+3) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
From Antti Karttunen, Dec 19 2015: (Start)
a(0)=a(1)=1; after which, for even n: a(n) = (n/2) * (n-1)!, and for odd n: a(n) = (n-1)/2 * ((n-1)! + (n-2)!). [The formula was empirically found after viewing these numbers in factorial base, A007623, and is easily proved by considering formulas from Lang (Apr 30 2010) and Detlefs (May 21 2010) shown above.]
For n >= 1, a(2*n+1) = a(2*n) + A153880(a(2*n)). [Follows from above.] (End)
Inverse Stirling transform of a(n) is (-1)^(n-1)*A009566(n). - Anton Zakharov, Aug 07 2016
a(n) ~ sqrt(Pi/2)*n^(n+1/2)/exp(n). - Ilya Gutkovskiy, Aug 07 2016
a(n) = A006595(n-1)*n/A000124(n) for n>=2. - Anton Zakharov, Aug 23 2016
a(n) = A001563(n-1) - A001286(n-1) for n>=2. - Anton Zakharov, Sep 23 2016
From Peter Bala, May 24 2017: (Start)
The o.g.f. A(x) satisfies the Riccati equation x^2*A'(x) + (x - 1)*A(x) + 1 - x^2 = 0.
G.f.: A(x) = 1 + x + x^2/(1 - 3*x/(1 - x/(1 - 4*x/(1 - 2*x/(1 - 5*x/(1 - 3*x/(1  - ... - (n + 2)*x/(1 - n*x/(1 - ... ))))))))) (apply Stokes, 1982).
A(x) = 1 + x + x^2/(1 - 2*x - x/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 3*x/(1 - 5*x/(1 - ... - n*x/(1 - (n+2)*x/(1 - ... ))))))))). (End)
H(x) = (1 - (1 + x)^(-2)) / 2 = x - 3*x^2/2! + 12*x^3/3! - ..., an e.g.f. for the signed sequence here (n!/2!), ignoring the first two terms, is the compositional inverse of G(x) = (1 - 2*x)^(-1/2) - 1 = x + 3*x^2/2! + 15*x^3/3! + ..., an e.g.f. for A001147. Cf. A094638. H(x) is the e.g.f. for the sequence (-1)^m * m!/2 for m = 2,3,4,... . Cf. A001715 for n!/3! and A001720 for n!/4!. Cf. columns of A094587, A173333, and A213936 and rows of A138533. - Tom Copeland, Dec 27 2019
From Amiram Eldar, Jan 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 2*(e-1).
Sum_{n>=0} (-1)^n/a(n) = 2/e. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Aug 20 2001

Further terms from Simone Severini, Oct 15 2004

cp's OEIS Frontend

A120299 Largest prime factor of Stirling numbers of first kind s(n,2) = A000254(n).

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A091826 (1/p)*(1+A000254(p)-p) as p runs through the primes.

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A091827 Least k such that k^n divides A000254(k).

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A091828 a(n)=n-2*valuation(A000254(n),3).

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A381681 a(n) is one of two integer components (with A000254) used in computing the inverse second moment of X+n, where X~Poisson(1).

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A010051 Characteristic function of primes: 1 if n is prime, else 0.

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A001008 a(n) = numerator of harmonic number H(n) = Sum_{i=1..n} 1/i.

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