A002162 -id:A002162 - cp's OEIS frontend

A058313 Numerator of the n-th alternating harmonic number, Sum_{k=1..n} (-1)^(k+1)/k.

1, 1, 5, 7, 47, 37, 319, 533, 1879, 1627, 20417, 18107, 263111, 237371, 52279, 95549, 1768477, 1632341, 33464927, 155685007, 166770367, 156188887, 3825136961, 3602044091, 19081066231, 18051406831, 57128792093, 7751493599, 236266661971

Offset: 1

N. J. A. Sloane, Dec 09 2000

A Wolstenholme-like theorem: for prime p > 3, if p = 6k-1, then p divides a(4k-1), otherwise if p = 6k+1, then p divides a(4k). - T. D. Noe, Apr 01 2004
For the limit n -> infinity of the partial sums of the alternating harmonic series see A002162. - Wolfdieter Lang, Sep 08 2015
a(n)/A058312(n) appears in the locker puzzle (see the links in A364317) as the probability of failures with the strategy used for n lockers and opening of up to floor(n/2) lockers. Note the alternative formula given below for a(n)/A058312(n) using only positive fractions. - Wolfdieter Lang, Aug 12 2023

			1, 1/2, 5/6, 7/12, 47/60, 37/60, 319/420, 533/840, 1879/2520, ...
For n=4: a(n)/A058312(n) = 7/12 because 1/1 - 1/2 + 1/3 - 1/4 = 7/12 = 1/4 + 1/3. - _Wolfdieter Lang_, Aug 12 2023

L. B. W. Jolley, Summation of Series, Dover Publications, 1961, page 14, #71.

T. D. Noe and Robert Israel, Table of n, a(n) for n = 1..2000 (terms up to a(200) from T. D. Noe)
Khristo N. Boyadzhiev, Power series with skew-harmonic numbers, dilogarithms, and double integrals, Tatra Mt. Math. Publ., Vol. 56 (2013), pp. 93-108.
Hisanori Mishima, Factorizations of many number sequences.
Hisanori Mishima, Factorizations of many number sequences.
Michael Penn, A number theory problem from an unlikely source, YouTube video, 2025.
Eric Weisstein's World of Mathematics, Harmonic Number.

Denominators are A058312. Cf. A025530.
Apart from leading term, same as A075830.
Cf. A001008 (numerator of n-th harmonic number).
Bisections are A049281 and A082687.
Cf. A181983.

import Data.Ratio((%), numerator)
a058313 n = a058313_list !! (n-1)
a058313_list = map numerator $ scanl1 (+) $ map (1 %) $ tail a181983_list
-- Reinhard Zumkeller, Mar 20 2013

A058313 := n->numer(add((-1)^(k+1)/k,k=1..n));
# Alternatively:
a := n -> numer(harmonic(n) - harmonic((n-modp(n,2))/2)):
seq(a(n), n=1..29); # Peter Luschny, May 03 2016

Numerator[Table[Sum[(-1)^(k + 1)/k, {k, n}], {n, 30}]] (* Harvey P. Dale, Jul 18 2012 *)
a[n_]:= (-1)^n (HarmonicNumber[n/2 - 1/2] - HarmonicNumber[n/2] + (-1)^n Log[4])/2; Table[a[n] // FullSimplify, {n, 29}] // Numerator (* Gerry Martens, Jul 05 2015 *)
Rest[Numerator[CoefficientList[Series[Log[1 + x]/(1 - x), {x, 0, 33}], x]]] (* Vincenzo Librandi, Jul 06 2015 *)
Table[Log[2] - (-1)^n LerchPhi[-1, 1, n + 1], {n, 20}] // Numerator (* Eric W. Weisstein, Aug 25 2023 *)

a(n)=(-1)^n*numerator(polcoeff(log(1-x)/(x+1)+O(x^(n+1)), n))

G.f. for a(n)/A058312(n): log(1+x)/(1-x). - Benoit Cloitre, Jun 15 2003
a(n) = (n*a(n-1) + (-1)^(n+1)*A058312(n-1))/gcd(n*a(n-1) + (-1)^(n+1)*A058312(n-1), n*A058312(n-1)). - Robert Israel, Jul 06 2015
From Peter Luschny, May 03 2016: (Start)
Let H(n) denote the harmonic numbers, AH(n) denote the alternating harmonic numbers, Psi the polygamma function and euler(n,x) the Euler polynomials. Then:
AH(n) = H(n) - H((n - n mod 2)/2).
AH(z) = log(2)+(1/2)*cos(Pi*z)*(Psi(z/2+1/2)-Psi(z/2+1)).
AH(z) ~ log(2)+(1/2)*cos(Pi*z)*(-1/z+1/(2*z^2)-1/(4*z^4)+1/(2*z^6)-...).
AH(z) ~ log(2)-(1/2)*cos(Pi*z)*Sum_{n>=0} Euler(n,0)/z^(n+1). (End)
Sum_{k>=1} (-1)^(k+1)*AH(k)/k = Pi^2/12 + log(2)^2/2 (Boyadzhiev, 2013). - Amiram Eldar, Oct 04 2021
a(n)/A058312(n) = Sum_{j=0..ceiling(n/2) - 1} 1/(n-j), for n >= 1. (Proof by comparing the recurrences for even and odd n.) - Wolfdieter Lang, Aug 12 2023
For n >= 1, log(2) = a(n)/A058312(n) + (-1)^n*n!*Sum_{k >= 1} 1/(k*(k + 1)* ...*(k + n)*2^k). - Peter Bala, Dec 07 2023
a(n) = the (reduced) numerator of the continued fraction 1/(1 + 1^2/(1 + 2^2/(1 + 3^2/(1 + ... + (n-1)^2/(1))))). - Peter Bala, Feb 18 2024

A004442 Natural numbers, pairs reversed: a(n) = n + (-1)^n; also Nimsum n + 1.

Original entry on oeis.org

1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14, 17, 16, 19, 18, 21, 20, 23, 22, 25, 24, 27, 26, 29, 28, 31, 30, 33, 32, 35, 34, 37, 36, 39, 38, 41, 40, 43, 42, 45, 44, 47, 46, 49, 48, 51, 50, 53, 52, 55, 54, 57, 56, 59, 58, 61, 60, 63, 62, 65, 64, 67, 66, 69

Offset: 0

N. J. A. Sloane

A self-inverse permutation of the natural numbers.
Nonnegative numbers rearranged with least disturbance to maintain a(n) not equal to n. - Amarnath Murthy, Sep 13 2002
Essentially lodumo_2 of A059841. - Philippe Deléham, Apr 26 2009
a(n) = A180176(n) for n >= 20. - Reinhard Zumkeller, Aug 15 2010

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Index entries for sequences related to Nim-sums
Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A003987, A004443, A004444. Equals A014681 - 1.
Cf. A005843, A005408, A059841.
Cf. A002162

import Data.List (transpose)
import Data.Bits (xor)
a004442 = xor 1 :: Integer -> Integer
a004442_list = concat $ transpose [a005408_list, a005843_list]
-- Reinhard Zumkeller, Jun 23 2013, Feb 01 2013, Oct 20 2011

a[0]:=1:a[1]:=0:for n from 2 to 70 do a[n]:=a[n-2]+2 od: seq(a[n], n=0..68); # Zerinvary Lajos, Feb 19 2008

Table[n + (-1)^n, {n, 0, 72}] (* or *)
CoefficientList[Series[(1 - x + 2x^2)/((1 - x)(1 - x^2)), {x, 0, 72}], x] (* Robert G. Wilson v, Jun 16 2006 *)
Flatten[Reverse/@Partition[Range[0,69],2]] (* or *) LinearRecurrence[{1,1,-1},{1,0,3},70] (* Harvey P. Dale, Jul 29 2018 *)

a(n)=n+(-1)^n \\ Charles R Greathouse IV, Nov 20 2012

Vec((1-x+2*x^2)/((1-x)*(1-x^2)) + O(x^100)) \\ Altug Alkan, Feb 04 2016

def a(n): return n^1
print([a(n) for n in range(69)]) # Michael S. Branicky, Jan 23 2022

a(n) = n XOR 1. - Odimar Fabeny, Sep 05 2004
G.f.: (1-x+2x^2)/((1-x)*(1-x^2)). - Mitchell Harris, Jan 10 2005
a(n+1) = lod_2(A059841(n)). - Philippe Deléham, Apr 26 2009
a(n) = 2*n - a(n-1) - 1 with n > 0, a(0)=1. - Vincenzo Librandi, Nov 18 2010
a(n) = Sum_{k=1..n-1} (-1)^(n-1-k)*C(n+1,k). - Mircea Merca, Feb 07 2013
For n > 1, a(n)^a(n) == 1 (mod n). - Thomas Ordowski, Jan 04 2016
Sum_{n>=0,n<>1} (-1)^n/a(n) = log(2) = A002162. - Peter McNair, Aug 07 2023

Offset adjusted by Reinhard Zumkeller, Mar 05 2010

A002266 Integers repeated 5 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16

Offset: 0

N. J. A. Sloane

For n > 3, number of consecutive "11's" after the (n+3) "1's" in the continued fraction for sqrt(L(n+2)/L(n)) where L(n) is the n-th Lucas number A000032 (see example). E.g., the continued fraction for sqrt(L(11)/L(9)) is [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 11, 58, 2, 4, 1, ...] with 12 consecutive ones followed by floor(11/5)=2 elevens. - Benoit Cloitre, Jan 08 2006
Complement of A010874, since A010874(n) + 5*a(n) = n. - Hieronymus Fischer, Jun 01 2007
From Paul Curtz, May 13 2020: (Start)
Main N-S vertical of the pentagonal spiral built with this sequence is A001105:
                              21
                          20  15  15
                      20  14  10  10  15
                  20  14   9   6   6  10  15
              20  14   9   5   3   3   6  10  15
          20  14   9   5   2   1   1   3   6  10  16
      19  14   9   5   2   0   0   0   1   3   6  11  16
        19   13   9   5  2   0   0    1   3   7 11  16
          19  13    8  5   2   2   1   4   7  11  16
            19   13   8  4   4   4   4   7  11  16
              19  13   8   8   8   7   7  11  17
                 18  13  12 12  12  12  12  17
                   18  18 18  18  17  17  17
The main S-N vertical and the next one are A000217. (End)

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

a(n) = A010766(n, 5).
Cf. A002162, A008648, A004526, A002264, A002265, A010761, A010762, A110532, A110533, A010872, A010873, A010874.
Partial sums: A130520.

a002266 = (`div` 5)
a002266_list = [0,0,0,0,0] ++ map (+ 1) a002266_list
-- Reinhard Zumkeller, Nov 27 2012

A002266:=n->floor(n/5); seq(A002266(n), n=0..100); # Wesley Ivan Hurt, Dec 10 2013

Table[Floor[n/5], {n,0,20}] (* Wesley Ivan Hurt, Dec 10 2013 *)
Table[{n,n,n,n,n},{n,0,20}]//Flatten (* Harvey P. Dale, Jun 17 2022 *)

a(n)=n\5 \\ Charles R Greathouse IV, Dec 10 2013

def A002266(n): return n//5 # Chai Wah Wu, Nov 08 2022

[floor(n/5) - 1 for n in range(5,88)] # Zerinvary Lajos, Dec 01 2009

a(n) = floor(n/5), n >= 0.
G.f.: x^5/((1-x)(1-x^5)).
a(n) = (n - A010874(n))/5. - Hieronymus Fischer, May 29 2007
For n >= 5, a(n) = floor(log_5(5^a(n-1) + 5^a(n-2) + 5^a(n-3) + 5^a(n-4) + 5^a(n-5))). - Vladimir Shevelev, Jun 22 2010
Sum_{n>=5} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, Sep 30 2022

Incorrect formula removed by Ridouane Oudra, Oct 16 2021

A003059 k appears 2k-1 times. Also, square root of n, rounded up.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

N. J. A. Sloane

n+1 first appears in the sequence at the A002522(n)-th entry (since the ultimate occurrence of n is n^2). a(n) refers to the greatest minimal length of monotone subsequence (i.e.either increasing or decreasing) contained within any sequence of n distinct numbers,according to the Erdős-Szekeres theorem. - Lekraj Beedassy, May 20 2003
With offset 0, apparently the least k such that binomial(2n,n-k) < (1/e) binomial(2n,n). - T. D. Noe, Mar 12 2009
a(n) is the number of nonnegative integer solutions of equation x + y^2 = n - 1. - Ran Pan, Oct 02 2015
Also the burning number of the cycle graph C_n (for n >= 4) and the path graph (for n >= 1). - Eric W. Weisstein, Jan 10 2024

T. D. Noe, Table of n, a(n) for n = 1..1000
Michael Somos, Sequences used for indexing triangular or square arrays.
Eric Weisstein's World of Mathematics, Burning Number
Eric Weisstein's World of Mathematics, Cycle Graph
Eric Weisstein's World of Mathematics, Path Graph

Cf. A000196, A000290, A002162, A002522, A010052, A157466.

a003059 n = a003059_list !! (n-1)
a003059_list = concat $ zipWith ($) (map replicate [1,3..]) [1..]
-- Reinhard Zumkeller, Mar 18 2011

[Ceiling(Sqrt(n)): n in [1..100]]; // G. C. Greubel, Nov 14 2018

A003059:=n->ceil(sqrt(n)); seq(A003059(k), k=1..100); # Wesley Ivan Hurt, Nov 08 2013

Table[ Table[n, {2n - 1}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jun 10 2013 *)
Ceiling[Sqrt[Range[100]]] (* G. C. Greubel, Nov 14 2018 *)
Table[PadRight[{},2k-1,k],{k,10}]//Flatten (* Harvey P. Dale, Jun 07 2020 *)

PARI
```
a(n)=if(n<1,0,1+sqrtint(n-1))
```

from math import isqrt
def A003059(n): return isqrt(n-1)+1 # Chai Wah Wu, Nov 14 2022

[ceil(sqrt(n)) for n in (1..100)] # G. C. Greubel, Nov 14 2018

a(n) = ceiling(sqrt(n)).
G.f.: (Sum_{n>=0} x^(n^2)) * x/(1-x). - Michael Somos, May 03 2003
a(n) = Sum_{k=0..n-1} A010052(k). - Reinhard Zumkeller, Mar 01 2009
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, Sep 29 2022

Name edited by M. F. Hasler, Nov 13 2018

A016627 Decimal expansion of log(4).

Original entry on oeis.org

1, 3, 8, 6, 2, 9, 4, 3, 6, 1, 1, 1, 9, 8, 9, 0, 6, 1, 8, 8, 3, 4, 4, 6, 4, 2, 4, 2, 9, 1, 6, 3, 5, 3, 1, 3, 6, 1, 5, 1, 0, 0, 0, 2, 6, 8, 7, 2, 0, 5, 1, 0, 5, 0, 8, 2, 4, 1, 3, 6, 0, 0, 1, 8, 9, 8, 6, 7, 8, 7, 2, 4, 3, 9, 3, 9, 3, 8, 9, 4, 3, 1, 2, 1, 1, 7, 2, 6, 6, 5, 3, 9, 9, 2, 8, 3, 7, 3, 7

Offset: 1

N. J. A. Sloane

This constant (negated) is the 1-dimensional analog of Madelung's constant. - Jean-François Alcover, May 20 2014
This constant is the sum over the reciprocals of the hexagonal numbers A000384(n), n >= 1. See the Downey et al. link, and the formula by Robert G. Wilson v below. - Wolfdieter Lang, Sep 12 2016
log(4) - 1 is the mean ratio between the smaller length and the larger length of the two parts of a stick that is being broken at a point that is uniformly chosen at random (Mosteller, 1965). - Amiram Eldar, Jul 25 2020
From Bernard Schott, Sep 11 2020: (Start)
This constant was the subject of the problem B5 during the 42nd Putnam competition in 1981 (see formula Sep 11 2020 and Putnam link).
Jeffrey Shallit generalizes this result obtained for base 2 to any base b (see Amer. Math. Month. link): Sum_{k>=1} digsum(k)_b / (k*(k+1)) = (b/(b-1)) * log(b) where digsum(k)_b is the sum of the digits of k when expressed in base b (for base 10 see A334388). (End)

			1.38629436111989061883446424291635313615100026872051050824136...

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
Frederick Mosteller, Fifty challenging problems of probability, Dover, New York, 1965. See problem 42, pp. 10 and 63.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:13:8 at page 23.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Lawrence Downey, Boon W. Ong, and James A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J., 39, no. 5 (2008), 391-394.
I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products (6th ed.), 2000, (eq. 0.236.1 and 0.236.6).
D. H. Lehmer, Interesting series involving the Central Binomial Coefficient, Am. Math. Monthly, Vol. 92, No. 7 (1985) pp. 449-457.
Allon G. Percus, Gabriel Istrate, Bruno Goncalves, Robert Z. Sumi, and Stefan Boettcher, The Peculiar Phase Structure of Random Graph Bisection, arXiv:0808.1549 [cond-mat.stat-mech], 2008.
H.-J. Seiffert, Problem B-771, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 32, No. 4 (1994), p. 374; More Sums, Solution to Problem B-771 by Don Redmond, ibid., Vol. 33, No. 5 (1995), pp. 470-471.
J. O. Shallit, Solutions of Advanced Problems, 6450, The American Mathematical Monthly, Vol. 92, No. 7, Aug.-Sep., 1985, pp. 513-514; DOI: 10.2307/2322523.
42nd Putnam Competition, Problem B5, 1981.
Index entries for sequences related to Olympiads and other Mathematical competitions.
Index entries for transcendental numbers.

Cf. A016732 (continued fraction).
Cf. A002162 (half), A133362 (reciprocal).
Cf. A000384, A001787, A002378, A007531, A066066, A069072, A191907.
Cf. A000045, A000120, A334388.

RealDigits[Log@ 4, 10, 111][[1]] (* Robert G. Wilson v, Aug 31 2014 *)

default(realprecision, 20080); x=log(4); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016627.txt", n, " ", d)); \\ Harry J. Smith, May 16 2009, corrected May 19 2009

A016627_vec(N)=digits(floor(log(precision(4.,N))*10^(N-1))) \\ Or: default(realprecision,N);digits(log(4)\.1^N) \\ M. F. Hasler, Oct 20 2013

log(4) = Sum_{k >= 1} H(k)/2^k where H(k) is the k-th harmonic number. - Benoit Cloitre, Jun 15 2003
Equals 1 - Sum_{k >= 1} (-1)^k/A002378(k) = 1 + 2*Sum_{k >= 0} 1/A069072(k) = 5/4 - Sum_{k >= 1} (-1)^k/A007531(k+2). - R. J. Mathar, Jan 23 2009
Equals 2*A002162 = Sum_{n >= 1} binomial(2*n, n)/(n*4^n) [D. H. Lehmer, Am. Math. Monthly 92 (1985) 449 and Jolley eq. 262]. - R. J. Mathar, Mar 04 2009
log(4) = Sum_{k >= 1} A191907(4, k)/k, (conjecture). - Mats Granvik, Jun 19 2011
log(4) = lim_{n -> infinity} A066066(n)/n. - M. F. Hasler, Oct 20 2013
Equals Sum_{k >= 1} 1/( 2*k^2 - k ). - Robert G. Wilson v, Aug 31 2014
Equals gamma(0, 1/2) - gamma(0, 1) = -(EulerGamma + polygamma(0, 1/2)), where gamma(n,x) denotes the generalized Stieltjes constants, see A020759. - Peter Luschny, May 16 2018
From Amiram Eldar, Jul 25 2020: (Start)
Equals Sum_{k>=1} (3/4)^k/k.
Equals Sum_{k>=1} 1/(k*2^(k-1)) = Sum_{k>=1} 1/A001787(k).
Equals Integral_{x=0..1} log(1+1/x) dx. (End)
Equals Sum_{k>=1} A000120(k) / (k*(k+1)). - Bernard Schott, Sep 11 2020
Equals 1 + Sum_{k>=1} zeta(2*k+1)/4^k. - Amiram Eldar, May 27 2021
Equals Sum_{k>=1} (2*k+1)*Fibonacci(k)/(k*(k+1)*2^k) (Seiffert, 1994). - Amiram Eldar, Jan 15 2022
Continued fraction: log(4) = 1 + 1/(2 + (1*2)/(2 + (2*3)/(2 + (3*4)/(2 + (4*5)/(2 + ... ))))) due to Euler. - Peter Bala, Mar 05 2024
log(4) = 2*Sum_{k>=1} 1/(k*P(k, 5/3)*P(k-1, 5/3)), where P(k, x) denotes the k-th Legendre polynomial. The first 20 terms of the series gives log(4) correct to 18 decimal places. - Peter Bala, Mar 18 2024
Equals Sum_{k>=1} (2*k - 1)!!/(k*(2*k)!!) [Ross] (see Spanier at p. 23). - Stefano Spezia, Dec 27 2024
Equals 1 + Sum_{k>=1} 1/(k*(4*k^2-1)). - Sean A. Irvine, Apr 05 2025
Equals Sum_{k>=1} (12*k^2-1)/(k*(4*k^2-1)^2). - Sean A. Irvine, Apr 06 2025
Equals Integral_{x=0..1} arctanh(sqrt(x))/sqrt(x) dx. - Kritsada Moomuang, Jun 06 2025
From Kritsada Moomuang, Jun 18 2025: (Start)
Equals Integral_{x=0..1} (x^(n - 1)*(x^(3*n) - 1))/log(x) dx, for n > 0.
Equals Integral_{x=0..Pi} sin(x)/(1 + abs(cos(x))) dx. (End)

A002392 Decimal expansion of natural logarithm of 10.

Original entry on oeis.org

2, 3, 0, 2, 5, 8, 5, 0, 9, 2, 9, 9, 4, 0, 4, 5, 6, 8, 4, 0, 1, 7, 9, 9, 1, 4, 5, 4, 6, 8, 4, 3, 6, 4, 2, 0, 7, 6, 0, 1, 1, 0, 1, 4, 8, 8, 6, 2, 8, 7, 7, 2, 9, 7, 6, 0, 3, 3, 3, 2, 7, 9, 0, 0, 9, 6, 7, 5, 7, 2, 6, 0, 9, 6, 7, 7, 3, 5, 2, 4, 8, 0, 2, 3, 5, 9, 9

Offset: 1

N. J. A. Sloane

			2.302585092994045684017991454684364207601101488628772976033327900967572...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 143.
W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. 2.
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).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 25, pages 227, 232.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 45.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Bakir Farhi, A curious result related to Kempner's series, arXiv:0807.3518 [math.NT], Jul 22 2008.
A. J. Kempner, A curious convergent series, Amer. Math. Monthly 23 (1914) pp. 48-50.
Simon Plouffe, log(10) the natural logarithm of 10 to 2000 digits.
Simon Plouffe, Plouffe's Inverter, The natural logarithm of 10 to 2000 digits.
Horace S. Uhler, Recalculation and extension of the modulus and of the logarithms of 2, 3, 5, 7 and 17, Proc. Nat. Acad. Sci. U. S. A. 26, (1940) pp. 205-212.
Eric Weisstein's World of Mathematics, Natural Logarithm of 10.
Index entries for transcendental numbers.

Cf. A016738 (continued fraction).

RealDigits[Log[10],10,120][[1]] (* Harvey P. Dale, Nov 23 2013 *)

default(realprecision, 20080); x=log(10); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002392.txt", n, " ", d)); \\ Harry J. Smith, Apr 16 2009

Equals A002162 + A016628. - R. J. Mathar, Jul 22 2025

A004524 Three even followed by one odd.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8, 8, 8, 9, 10, 10, 10, 11, 12, 12, 12, 13, 14, 14, 14, 15, 16, 16, 16, 17, 18, 18, 18, 19, 20, 20, 20, 21, 22, 22, 22, 23, 24, 24, 24, 25, 26, 26, 26, 27, 28, 28, 28, 29, 30, 30, 30, 31, 32, 32, 32, 33, 34, 34, 34, 35, 36, 36, 36, 37

Offset: 0

N. J. A. Sloane

Ignoring the first term, for n >= 0, n/2 rounded by the method called "banker's rounding", "statistician's rounding", or "round-to-even" gives 0, 0, 1, 2, 2, 2, 3, ..., where this method rounds k + 0.5 to k if positive integer k is even but rounds k + 0.5 to k + 1 when k + 1 is even. (If the method is indeed defined such that the above statement is also true with the word "positive" removed, then the first 0 term need not be ignored and this sequence can be further extended symmetrically with a(m) = -a(-m) for all integers m, an advantage over usual rounding.) The corresponding sequence for n/2 rounded by the common method is A004526 (considered as beginning with n = -1). - Rick L. Shepherd, Nov 16 2006
From Anthony Hernandez, Aug 08 2016: (Start)
Arrange the positive integers starting at 1 into a triangular array
   1
   2  3
   4  5  6
   7  8  9 10
  11 12 13 14 15
  16 17 18 19 20 21
  22 23 24 25 26 27 28
  29 30 31 32 33 34 35 36
and let e(n) count the even numbers in the n-th row of the array. Then e(n) = a(n+1). For example, e(6) = a(7) = 3 and there are three even numbers in the 6th row of the array. For the count of odd numbers, f(n), look at the sequence A004525. (End)
Also the domination number of the (n-1) X (n-1) white bishop graph. - Eric W. Weisstein, Jun 26 2017
Let (b(n)) be the p-INVERT of A010892 using p(S) = 1 - S^2; then b(n) = a(n+1) for n >= 0. See A292301. - Clark Kimberling, Sep 30 2017
Also the total domination number of the (n-2)-complete graph (for n>3), (n-2)-cycle graph (for n>4), and (n-2)-pan graph (for n>4). - Eric W. Weisstein, Apr 07 2018
The sequence is the interleaving of the duplicated even integers (A052928) with the nonnegative integers (A001477). - Guenther Schrack, Mar 05 2019

			G.f. = x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 4*x^9 + 4*x^10 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Charles H. Conley and Valentin Ovsienko, Shadows of rationals and irrationals: supersymmetric continued fractions and the super modular group, arXiv:2209.10426 [math-ph], 2022.
David Cushing, Stuart Gipp, Ezra Levick, Em Rickinson, and David I. Stewart, Optimal play in Guess Who, arXiv:2508.00799 [math.CO], 2025. See p. 3.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
Eric Weisstein's World of Mathematics, Complete Graph.
Eric Weisstein's World of Mathematics, Cycle Graph.
Eric Weisstein's World of Mathematics, Domination Number.
Eric Weisstein's World of Mathematics, Pan Graph.
Eric Weisstein's World of Mathematics, Total Domination Number.
Eric Weisstein's World of Mathematics, White Bishop Graph.
Wikipedia, Rounding.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
Index entries for two-way infinite sequences.

Cf. A002162, A001477, A004525, A004526, A010892, A052928, A092038, A093390, A093391, A093392, A093393, A292301.
Zero followed by partial sums of A021913.
First differences of A011848.

List([0..79],n->Int(n/4)+Int((n+1)/4)); # Muniru A Asiru, Mar 06 2019

a004524 n = n `div` 4 + (n + 1) `div` 4
a004524_list = 0 : 0 : 0 : 1 : map (+ 2) a004524_list
-- Reinhard Zumkeller, Feb 22 2013, Jul 14 2012

[Floor(n/4)+Floor((n+1)/4) : n in [0..80]]; // Wesley Ivan Hurt, Jul 21 2014

A004524:=n->floor(n/4)+floor((n+1)/4): seq(A004524(n), n=0..50); # Wesley Ivan Hurt, Jul 21 2014

Table[Floor[n/4] + Floor[(n + 1)/4], {n, 0, 80}] (* Wesley Ivan Hurt, Jul 21 2014 *)
Flatten[Table[{n, n, n, n + 1}, {n, 0, 38, 2}]] (* Alonso del Arte, Aug 10 2016 *)
Table[(n + Cos[n Pi/2] - 1)/2, {n, 0, 80}] (* Eric W. Weisstein, Apr 07 2018 *)
Table[Floor[n/2 - 1] + Ceiling[n/4 - 1/2] - Floor[n/4 - 1/2], {n, 0, 80}] (* Eric W. Weisstein, Apr 07 2018 *)
LinearRecurrence[{2, -2, 2, -1}, {0, 0, 1, 2}, {0, 80}] (* Eric W. Weisstein, Apr 07 2018 *)
CoefficientList[Series[x^3/((1 - x)^2 (1 + x^2)), {x, 0, 80}], x] (* Eric W. Weisstein, Apr 07 2018 *)
Table[Round[(n - 1)/2], {n, 0, 20}] (* Eric W. Weisstein, Jun 19 2024 *)
Round[(Range[0, 20] - 1)/2] (* Eric W. Weisstein, Jun 19 2024 *)
Table[PadRight[{},If[EvenQ[n],3,1],n],{n,0,40}]//Flatten (* Harvey P. Dale, Dec 11 2024 *)

{a(n) = n\4 + (n+1)\4}; /* Michael Somos, Jul 19 2003 */

concat([0,0,0], Vec(x^3/((1-x)^2*(1+x^2)) + O(x^80))) \\ Altug Alkan, Oct 31 2015

def A004524(n): return (n>>2)+(n+1>>2) # Chai Wah Wu, Jul 29 2022

[floor(n/4)+floor((n+1)/4) for n in (0..80)] # G. C. Greubel, Mar 08 2019

a(n) = a(n-1) - a(n-2) + a(n-3) + 1 = (n-1) - A004525(n-1). - Henry Bottomley, Mar 08 2000
G.f.: x^3/((1 - x)^2*(1 + x^2)) = x^3*(1 - x^2)/((1 - x)^2*(1 - x^4)). - Michael Somos, Jul 19 2003
If the sequence is extended to negative arguments in the natural way, it satisfies a(n) = -a(2-n) for all n in Z. - Michael Somos, Jul 19 2003
a(n) = A092038(n-3) for n > 4. - Reinhard Zumkeller, Mar 28 2004
From Paul Barry, Oct 27 2004: (Start)
E.g.f.: (exp(x)*(x-1) + cos(x))/2.
a(n) = (n - 1 - cos(Pi*(n-2)/2))/2. (End)
a(n+3) = Sum_{k = 0..n} (1 + (-1)^C(n,2))/2. - Paul Barry, Mar 31 2008
a(n) = floor(n/4) + floor((n+1)/4). - Arkadiusz Wesolowski, Sep 19 2012
From Wesley Ivan Hurt, Jul 21 2014, Oct 31 2015: (Start)
a(n) = Sum_{i = 1..n-1} (floor(i/2) mod 2).
a(n) = n/2 - sqrt(n^2 mod 8)/2. (End)
Euler transform of length 4 sequence [2, -1, 0, 1]. - Michael Somos, Apr 03 2017
a(n) = (2*n - 2 + (1 + (-1)^n)*(-1)^(n*(n-1)/2))/4. - Guenther Schrack, Mar 04 2019
Sum_{n>=3} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, Sep 29 2022

A118324 (Greedy) Egyptian fraction expansion of log 2.

Original entry on oeis.org

2, 6, 38, 6071, 144715221, 58600453312405245, 28261174043083404192255923187258021, 1350299665604204277005894785275782053022737307184211775676631561245153

Offset: 1

Eric W. Weisstein, Apr 23 2006

			log(2) = 1/2 + 1/6 + 1/38 + 1/6071 + 1/144715221 + ...

Amiram Eldar, Table of n, a(n) for n = 1..11
Eric Weisstein's World of Mathematics, Egyptian Fraction.
Index entries for sequences related to Egyptian fractions.

Cf. A002162 (log(2)).

lst={};k=N[Log[2],1000];Do[s=Ceiling[1/k];AppendTo[lst,s];k=k-1/s,{n,12}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 02 2009 *)

x=log(2); for (k=1,8,d=ceil(1/x);x=x-1/d;print(d)) \\ Jaume Oliver Lafont, Feb 24 2009

a(8) from Jaume Oliver Lafont, Feb 24 2009

A132747 a(n) = number of non-isolated divisors of n.

Original entry on oeis.org

0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 3, 0, 4, 0, 2, 0, 4, 0, 2, 0, 2, 0, 5, 0, 2, 0, 2, 0, 4, 0, 2, 0, 4, 0, 5, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 3, 0, 4, 0, 2, 0, 6, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 6, 0, 2, 0, 2, 0, 3, 0, 4, 0, 2, 0, 6, 0, 2, 0, 2, 0, 7, 0, 2, 0, 2, 0, 4, 0, 2, 0, 4, 0, 3, 0, 2, 0

Offset: 1

Leroy Quet, Aug 27 2007

nonn

A divisor d of n is non-isolated if either d-1 or d+1 divides n. a(2n-1) = 0 for all n >= 1.

			The positive divisors of 20 are 1,2,4,5,10,20. Of these, 1 and 2 are next to each other and 4 and 5 are next to each other. So a(20) = the number of these divisors, which is 4.

Ray Chandler, Table of n, a(n) for n=1..10000

Cf. A000005, A002162, A129308, A132748, A132881.

Table[Length[Select[Divisors[n], If[ # > 1, IntegerQ[n/(#*(# - 1))]] || IntegerQ[n/(#*(# + 1))] &]], {n, 1, 90}] (* Stefan Steinerberger, Oct 26 2007 *)

a(n) = my(div = divisors(n)); sumdiv(n, d, vecsearch(div, d-1) || vecsearch(div, d+1)); \\ Michel Marcus, Aug 22 2014

a(n) = A000005(n) - A132881(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(2) + 1 = A002162 + 1 = 1.693147.... . - Amiram Eldar, Mar 22 2024

More terms from Stefan Steinerberger, Oct 26 2007

Extended by Ray Chandler, Jun 24 2008

A152901 Decimal expansion of log_24 (2).

Original entry on oeis.org

2, 1, 8, 1, 0, 4, 2, 9, 1, 9, 8, 5, 5, 3, 1, 5, 5, 9, 2, 2, 9, 3, 3, 7, 8, 0, 6, 4, 4, 3, 3, 8, 8, 3, 8, 8, 6, 2, 7, 6, 6, 0, 4, 2, 5, 4, 9, 9, 4, 9, 9, 7, 6, 7, 6, 1, 4, 7, 3, 8, 1, 5, 6, 4, 6, 4, 1, 8, 4, 2, 3, 8, 1, 5, 0, 0, 9, 9, 2, 3, 5, 0, 9, 7, 9, 9, 1, 7, 7, 1, 4, 1, 8, 8, 2, 2, 2, 1, 4

Offset: 0

N. J. A. Sloane, Oct 30 2009

			0.21810429198553155922933780644338838862766042549949976761473...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_24(m): this sequence, A153100 (m=3), A153200 (m=4), A153458 (m=5), A153614 (m=6), A153736 (m=7), A154007 (m=8), A154116 (m=9), A154174 (m=10), A154195 (m=11), A154216 (m=12), A154461 (m=13), A154538 (m=14), A154735 (m=15), A154846 (m=16), A154904 (m=17), A154994 (m=18), A155168 (m=19), A155535 (m=20), A155692 (m=21), A155792 (m=22), A155920 (m=23).

RealDigits[Log[24, 2], 10, 100][[1]] (* Vincenzo Librandi, Sep 10 2013 *)

log(2)/log(24) \\ Charles R Greathouse IV, Aug 06 2020

A002162 divided by A016647. - R. J. Mathar, Mar 22 2011

cp's OEIS Frontend

A058313 Numerator of the n-th alternating harmonic number, Sum_{k=1..n} (-1)^(k+1)/k.

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A004442 Natural numbers, pairs reversed: a(n) = n + (-1)^n; also Nimsum n + 1.

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A002266 Integers repeated 5 times.

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A003059 k appears 2k-1 times. Also, square root of n, rounded up.

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A016627 Decimal expansion of log(4).

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