A222171 -id:A222171 - cp's OEIS frontend

A005843 The nonnegative even numbers: a(n) = 2n.

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120

Offset: 0

N. J. A. Sloane

-2, -4, -6, -8, -10, -12, -14, ... are the trivial zeros of the Riemann zeta function. - Vivek Suri (vsuri(AT)jhu.edu), Jan 24 2008
If a 2-set Y and an (n-2)-set Z are disjoint subsets of an n-set X then a(n-2) is the number of 2-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 19 2007
A134452(a(n)) = 0; A134451(a(n)) = 2 for n > 0. - Reinhard Zumkeller, Oct 27 2007
Omitting the initial zero gives the number of prime divisors with multiplicity of product of terms of n-th row of A077553. - Ray Chandler, Aug 21 2003
A059841(a(n))=1, A000035(a(n))=0. - Reinhard Zumkeller, Sep 29 2008
(APSO) Alternating partial sums of (a-b+c-d+e-f+g...) = (a+b+c+d+e+f+g...) - 2*(b+d+f...), it appears that APSO(A005843) = A052928 = A002378 - 2*(A116471), with A116471=2*A008794. - Eric Desbiaux, Oct 28 2008
A056753(a(n)) = 1. - Reinhard Zumkeller, Aug 23 2009
Twice the nonnegative numbers. - Juri-Stepan Gerasimov, Dec 12 2009
The number of hydrogen atoms in straight-chain (C(n)H(2n+2)), branched (C(n)H(2n+2), n > 3), and cyclic, n-carbon alkanes (C(n)H(2n), n > 2). - Paul Muljadi, Feb 18 2010
For n >= 1; a(n) = the smallest numbers m with the number of steps n of iterations of {r - (smallest prime divisor of r)} needed to reach 0 starting at r = m. See A175126 and A175127. A175126(a(n)) = A175126(A175127(n)) = n. Example (a(4)=8): 8-2=6, 6-2=4, 4-2=2, 2-2=0; iterations has 4 steps and number 8 is the smallest number with such result. - Jaroslav Krizek, Feb 15 2010
For n >= 1, a(n) = numbers k such that arithmetic mean of the first k positive integers is not integer. A040001(a(n)) > 1. See A145051 and A040001. - Jaroslav Krizek, May 28 2010
Union of A179082 and A179083. - Reinhard Zumkeller, Jun 28 2010
a(k) is the (Moore lower bound on and the) order of the (k,4)-cage: the smallest k-regular graph having girth four: the complete bipartite graph with k vertices in each part. - Jason Kimberley, Oct 30 2011
For n > 0: A048272(a(n)) <= 0. - Reinhard Zumkeller, Jan 21 2012
Let n be the number of pancakes that have to be divided equally between n+1 children. a(n) is the minimal number of radial cuts needed to accomplish the task. - Ivan N. Ianakiev, Sep 18 2013
For n > 0, a(n) is the largest number k such that (k!-n)/(k-n) is an integer. - Derek Orr, Jul 02 2014
a(n) when n > 2 is also the number of permutations simultaneously avoiding 213, 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl Aug 07 2014
It appears that for n > 2, a(n) = A020482(n) + A002373(n), where all sequences are infinite. This is consistent with Goldbach's conjecture, which states that every even number > 2 can be expressed as the sum of two prime numbers. - Bob Selcoe, Mar 08 2015
Number of partitions of 4n into exactly 2 parts. - Colin Barker, Mar 23 2015
Number of neighbors in von Neumann neighborhood. - Dmitry Zaitsev, Nov 30 2015
Unique solution b( ) of the complementary equation a(n) = a(n-1)^2 - a(n-2)*b(n-1), where a(0) = 1, a(1) = 3, and a( ) and b( ) are increasing complementary sequences. - Clark Kimberling, Nov 21 2017
Also the maximum number of non-attacking bishops on an (n+1) X (n+1) board (n>0). (Cf. A000027 for rooks and queens (n>3), A008794 for kings or A030978 for knights.) - Martin Renner, Jan 26 2020
Integer k is even positive iff phi(2k) > phi(k), where phi is Euler's totient (A000010) [see reference De Koninck & Mercier]. - Bernard Schott, Dec 10 2020
Number of 3-permutations of n elements avoiding the patterns 132, 213, 312 and also number of 3-permutations avoiding the patterns 213, 231, 321. See Bonichon and Sun. - Michel Marcus, Aug 20 2022
a(n) gives the y-value of the integral solution (x,y) of the Pellian equation x^2 - (n^2 + 1)*y^2 = 1. The x-value is given by 2*n^2 + 1 (see Tattersall). - Stefano Spezia, Jul 24 2025

			G.f. = 2*x + 4*x^2 + 6*x^3 + 8*x^4 + 10*x^5 + 12*x^6 + 14*x^7 + 16*x^8 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 28.
J.-M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 529a pp. 71 and 257, Ellipses, 2004, Paris.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 256.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.
Charles Cratty, Samuel Erickson, Frehiwet Negass, and Lara Pudwell, Pattern Avoidance in Double Lists, preprint, 2015.
Kevin Fagan, Drabble cartoon, Jun 15 1987: Intelligence Test
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012, J. Int. Seq. 15 (2012) # 12.6.2
Milan Janjic, Two Enumerative Functions
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Even Number
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros
Wikipedia, Alkane
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

a(n)=2*A001477(n). - Juri-Stepan Gerasimov, Dec 12 2009
Cf. A000027, A002061, A005408, A001358, A077553, A077554, A077555, A002024, A087112, A157888, A157889, A140811, A157872, A157909, A157910, A165900.
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), this sequence (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Oct 30 2011
Cf. A231200 (boustrophedon transform).

a005843 = (* 2)
a005843_list = [0, 2 ..]  -- Reinhard Zumkeller, Feb 11 2012

Magma
```
[ 2*n : n in [0..100]];
```

A005843 := n->2*n;
A005843:=2/(z-1)**2; # Simon Plouffe in his 1992 dissertation

Range[0,120,2] (* Harvey P. Dale, Aug 16 2011 *)

PARI
```
A005843(n) = 2*n
```

def a(n): return 2*n # Martin Gergov, Oct 20 2022

R
```
seq(0,200,2)
```

G.f.: 2*x/(1-x)^2.
E.g.f.: 2*x*exp(x). - Geoffrey Critzer, Aug 25 2012
G.f. with interpolated zeros: 2x^2/((1-x)^2 * (1+x)^2); e.g.f. with interpolated zeros: x*sinh(x). - Geoffrey Critzer, Aug 25 2012
Inverse binomial transform of A036289, n*2^n. - Joshua Zucker, Jan 13 2006
a(0) = 0, a(1) = 2, a(n) = 2a(n-1) - a(n-2). - Jaume Oliver Lafont, May 07 2008
a(n) = Sum_{k=1..n} floor(6n/4^k + 1/2). - Vladimir Shevelev, Jun 04 2009
a(n) = A034856(n+1) - A000124(n) = A000217(n) + A005408(n) - A000124(n) = A005408(n) - 1. - Jaroslav Krizek, Sep 05 2009
a(n) = Sum_{k>=0} A030308(n,k)*A000079(k+1). - Philippe Deléham, Oct 17 2011
Digit sequence 22 read in base n-1. - Jason Kimberley, Oct 30 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Dec 23 2011
a(n) = 2*n = Product_{k=1..2*n-1} 2*sin(Pi*k/(2*n)), n >= 0 (undefined product := 1). See an Oct 09 2013 formula contribution in A000027 with a reference. - Wolfdieter Lang, Oct 10 2013
From Ilya Gutkovskiy, Aug 19 2016: (Start)
Convolution of A007395 and A057427.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2 = (1/2)*A002162 = (1/10)*A016655. (End)
From Bernard Schott, Dec 10 2020: (Start)
Sum_{n>=1} 1/a(n)^2 = Pi^2/24 = A222171.
Sum_{n>=1} (-1)^(n+1)/a(n)^2 = Pi^2/48 = A245058. (End)

A001105 a(n) = 2*n^2.

Original entry on oeis.org

0, 2, 8, 18, 32, 50, 72, 98, 128, 162, 200, 242, 288, 338, 392, 450, 512, 578, 648, 722, 800, 882, 968, 1058, 1152, 1250, 1352, 1458, 1568, 1682, 1800, 1922, 2048, 2178, 2312, 2450, 2592, 2738, 2888, 3042, 3200, 3362, 3528, 3698, 3872, 4050, 4232, 4418

Offset: 0

Bernd.Walter(AT)frankfurt.netsurf.de

Number of edges of the complete bipartite graph of order 3n, K_{n,2n}. - Roberto E. Martinez II, Jan 07 2002
"If each period in the periodic system ends in a rare gas ..., the number of elements in a period can be found from the ordinal number n of the period by the formula: L = ((2n+3+(-1)^n)^2)/8..." - Nature, Jun 09 1951; Nature 411 (Jun 07 2001), p. 648. This produces the present sequence doubled up.
Let z(1) = i = sqrt(-1), z(k+1) = 1/(z(k)+2i); then a(n) = (-1)*Imag(z(n+1))/Real(z(n+1)). - Benoit Cloitre, Aug 06 2002
Maximum number of electrons in an atomic shell with total quantum number n. Partial sums of A016825. - Jeremy Gardiner, Dec 19 2004
Arithmetic mean of triangular numbers in pairs: (1+3)/2, (6+10)/2, (15+21)/2, ... . - Amarnath Murthy, Aug 05 2005
These numbers form a pattern on the Ulam spiral similar to that of the triangular numbers. - G. Roda, Oct 20 2010
Integral areas of isosceles right triangles with rational legs (legs are 2n and triangles are nondegenerate for n > 0). - Rick L. Shepherd, Sep 29 2009
Even squares divided by 2. - Omar E. Pol, Aug 18 2011
Number of stars when distributed as in the U.S.A. flag: n rows with n+1 stars and, between each pair of these, one row with n stars (i.e., n-1 of these), i.e., n*(n+1)+(n-1)*n = 2*n^2 = A001105(n). - César Eliud Lozada, Sep 17 2012
Apparently the number of Dyck paths with semilength n+3 and an odd number of peaks and the central peak having height n-3. - David Scambler, Apr 29 2013
Sum of the partition parts of 2n into exactly two parts. - Wesley Ivan Hurt, Jun 01 2013
Consider primitive Pythagorean triangles (a^2 + b^2 = c^2, gcd(a, b) = 1) with hypotenuse c (A020882) and respective odd leg a (A180620); sequence gives values c-a, sorted with duplicates removed. - K. G. Stier, Nov 04 2013
Number of roots in the root systems of type B_n and C_n (for n > 1). - Tom Edgar, Nov 05 2013
Area of a square with diagonal 2n. - Wesley Ivan Hurt, Jun 18 2014
This sequence appears also as the first and second member of the quartet [a(n), a(n), p(n), p(n)] of the square of [n, n, n+1, n+1] in the Clifford algebra Cl_2 for n >= 0. p(n) = A046092(n). See an Oct 15 2014 comment on A147973 where also a reference is given. - Wolfdieter Lang, Oct 16 2014
a(n) are the only integers m where (A000005(m) + A000203(m)) = (number of divisors of m + sum of divisors of m) is an odd number. - Richard R. Forberg, Jan 09 2015
a(n) represents the first term in a sum of consecutive integers running to a(n+1)-1 that equals (2n+1)^3. - Patrick J. McNab, Dec 24 2016
Also the number of 3-cycles in the (n+4)-triangular honeycomb obtuse knight graph. - Eric W. Weisstein, Jul 29 2017
Also the Wiener index of the n-cocktail party graph for n > 1. - Eric W. Weisstein, Sep 07 2017
Numbers represented as the palindrome 242 in number base B including B=2 (binary), 3 (ternary) and 4: 242(2)=18, 242(3)=32, 242(4)=50, ... 242(9)=200, 242(10)=242, ... - Ron Knott, Nov 14 2017
a(n) is the square of the hypotenuse of an isosceles right triangle whose sides are equal to n. - Thomas M. Green, Aug 20 2019
The sequence contains all odd powers of 2 (A004171) but no even power of 2 (A000302). - Torlach Rush, Oct 10 2019
From Bernard Schott, Aug 31 2021 and Sep 16 2021: (Start)
Apart from 0, integers such that the number of even divisors (A183063) is odd.
Proof: every n = 2^q * (2k+1), q, k >= 0, then 2*n^2 = 2^(2q+1) * (2k+1)^2; now, gcd(2, 2k+1) = 1, tau(2^(2q+1)) = 2q+2 and tau((2k+1)^2) = 2u+1 because (2k+1)^2 is square, so, tau(2*n^2) = (2q+2) * (2u+1).
The 2q+2 divisors of 2^(2q+1) are {1, 2, 2^2, 2^3, ..., 2^(2q+1)}, so 2^(2q+1) has 2q+1 even divisors {2^1, 2^2, 2^3, ..., 2^(2q+1)}.
Conclusion: these 2q+1 even divisors create with the 2u+1 odd divisors of (2k+1)^2 exactly (2q+1)*(2u+1) even divisors of 2*n^2, and (2q+1)*(2u+1) is odd. (End)
a(n) with n>0 are the numbers with period length 2 for Bulgarian and Mancala solitaire. - Paul Weisenhorn, Jan 29 2022
Number of points at L1 distance = 2 from any given point in Z^n. - Shel Kaphan, Feb 25 2023
Integer that multiplies (h^2)/(m*L^2) to give the energy of a 1-D quantum mechanical particle in a box whenever it is an integer multiple of (h^2)/(m*L^2), where h = Planck's constant, m = mass of particle, and L = length of box. - A. Timothy Royappa, Mar 14 2025

			a(3) = 18; since 2(3) = 6 has 3 partitions with exactly two parts: (5,1), (4,2), (3,3).  Adding all the parts, we get: 1 + 2 + 3 + 3 + 4 + 5 = 18. - _Wesley Ivan Hurt_, Jun 01 2013

Peter Atkins, Julio De Paula, and James Keeler, "Atkins' Physical Chemistry," Oxford University Press, 2023, p. 31.
Arthur Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
Martin Gardner, The Colossal Book of Mathematics, Classic Puzzles, Paradoxes and Problems, Chapter 2 entitled "The Calculus of Finite Differences," W. W. Norton and Company, New York, 2001, pages 12-13.
L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 44.
Alain M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000, p. 213.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Lancelot Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
Vladimir Ladma, Magic Numbers.
Vladimir Pletser, General solutions of sums of consecutive cubed integers equal to squared integers, arXiv:1501.06098 [math.NT], 2015.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A006331 (partial sums), A016742, A056106, A116471, A245508, A251599, A002061, A165900.
Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488.
Cf. A058331 and A247375. - Bruno Berselli, Sep 16 2014
Cf. A194715 (4-cycles in the triangular honeycomb obtuse knight graph), A290391 (5-cycles), A290392 (6-cycles). - Eric W. Weisstein, Jul 29 2017
Cf. A139098, A077591.
Cf. A000217, A002266.
Integers such that: this sequence (the number of even divisors is odd), A028982 (the number of odd divisors is odd), A028983 (the number of odd divisors is even), A183300 (the number of even divisors is even).

List([0..50],n->2*n^2); # Muniru A Asiru, Feb 24 2019

a001105 = a005843 . a000290  -- Reinhard Zumkeller, Dec 12 2012

[2*n^2: n in [0..50] ]; // Vincenzo Librandi, Apr 30 2011

A001105:=n->2*n^2; seq(A001105(k), k=0..100); # Wesley Ivan Hurt, Oct 29 2013

2 Range[0, 50]^2 (* Harvey P. Dale, Jan 23 2011 *)
LinearRecurrence[{3, -3, 1}, {2, 8, 18}, {0, 20}] (* Eric W. Weisstein, Jul 28 2017 *)
2 PolygonalNumber[4, Range[0, 20]] (* Eric W. Weisstein, Jul 28 2017 *)

a(n) = 2*n^2; \\ Charles R Greathouse IV, Jun 16 2011

def A001105(n): return n**2<<1 # Chai Wah Wu, Aug 04 2025

[2*n^2 for n in (0..20)] # G. C. Greubel, Feb 22 2019

a(n) = (-1)^(n+1) * A053120(2*n, 2).
G.f.: 2*x*(1+x)/(1-x)^3.
a(n) = A100345(n, n).
Sum_{n>=1} 1/a(n) = Pi^2/12 =A072691. [Jolley eq. 319]. - Gary W. Adamson, Dec 21 2006
a(n) = A049452(n) - A033991(n). - Zerinvary Lajos, Jun 12 2007
a(n) = A016742(n)/2. - Zerinvary Lajos, Jun 20 2008
a(n) = 2 * A000290(n). - Omar E. Pol, May 14 2008
a(n) = 4*n + a(n-1) - 2, n > 0. - Vincenzo Librandi
a(n) = A002378(n-1) + A002378(n). - Joerg M. Schuetze (joerg(AT)cyberheim.de), Mar 08 2010 [Corrected by Klaus Purath, Jun 18 2020]
a(n) = A176271(n,k) + A176271(n,n-k+1), 1 <= k <= n. - Reinhard Zumkeller, Apr 13 2010
a(n) = A007607(A000290(n)). - Reinhard Zumkeller, Feb 12 2011
For n > 0, a(n) = 1/coefficient of x^2 in the Maclaurin expansion of 1/(cos(x)+n-1). - Francesco Daddi, Aug 04 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Artur Jasinski, Nov 24 2011
a(n) = A070216(n,n) for n > 0. - Reinhard Zumkeller, Nov 11 2012
a(n) = A014132(2*n-1,n) for n > 0. - Reinhard Zumkeller, Dec 12 2012
a(n) = A000217(n) + A000326(n). - Omar E. Pol, Jan 11 2013
(a(n) - A000217(k))^2  = A000217(2*n-1-k)*A000217(2*n+k) + n^2, for all k. - Charlie Marion, May 04 2013
a(n) = floor(1/(1-cos(1/n))), n > 0. - Clark Kimberling, Oct 08 2014
a(n) = A251599(3*n-1) for n > 0. - Reinhard Zumkeller, Dec 13 2014
a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n+4)/3). - Wesley Ivan Hurt, Mar 12 2015
a(n) = A002061(n+1) + A165900(n). - Torlach Rush, Feb 21 2019
E.g.f.: 2*exp(x)*x*(1 + x). - Stefano Spezia, Oct 12 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/24 (A222171). - Amiram Eldar, Jul 03 2020
From Amiram Eldar, Feb 03 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = sqrt(2)*sinh(Pi/sqrt(2))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(2)*sin(Pi/sqrt(2))/Pi. (End)

A146076 Sum of even divisors of n.

Original entry on oeis.org

0, 2, 0, 6, 0, 8, 0, 14, 0, 12, 0, 24, 0, 16, 0, 30, 0, 26, 0, 36, 0, 24, 0, 56, 0, 28, 0, 48, 0, 48, 0, 62, 0, 36, 0, 78, 0, 40, 0, 84, 0, 64, 0, 72, 0, 48, 0, 120, 0, 62, 0, 84, 0, 80, 0, 112, 0, 60, 0, 144, 0, 64, 0, 126, 0, 96, 0, 108, 0, 96, 0, 182, 0, 76, 0, 120, 0, 112, 0, 180, 0, 84, 0, 192, 0, 88, 0, 168, 0, 156

Offset: 1

N. J. A. Sloane, Apr 09 2009

The usual OEIS policy is not to include sequences like this where alternate terms are zero; this is an exception. A074400 is the main entry.

a(n) is also the total number of parts in all partitions of n into an even number of equal parts. - Omar E. Pol, Jun 04 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A000203, A000593, A006128, A038712, A065442, A074400, A183063, A222171.

A146076 := proc(n)
    if type(n,'even') then
        2*numtheory[sigma](n/2) ;
    else
        0;
    end if;
end proc: # R. J. Mathar, Dec 07 2017

f[n_] := Plus @@ Select[Divisors[n], EvenQ]; Array[f, 150] (* Vincenzo Librandi, May 17 2013 *)
a[n_] := DivisorSum[n, Boole[EvenQ[#]]*#&]; Array[a, 100] (* Jean-François Alcover, Dec 01 2015 *)
Table[CoefficientList[Series[-Log[QPochhammer[x^2, x^2]], {x, 0, 60}],x][[n + 1]] n, {n, 1, 60}] (* Benedict W. J. Irwin, Jul 04 2016 *)
a[n_] := If[OddQ[n], 0, 2*DivisorSigma[1, n/2]]; Array[a, 100] (* Amiram Eldar, Jan 11 2023 *)

vector(80, n, if (n%2, 0, sumdiv(n, d, d*(1-(d%2))))) \\ Michel Marcus, Mar 30 2015

a(n) = if (n%2, 0, 2*sigma(n/2)); \\ Michel Marcus, Apr 01 2015

a(2k-1) = 0, a(2k) = 2*sigma(k) for positive k.
Dirichlet g.f.: zeta(s - 1)*zeta(s)*2^(1 - s). - Geoffrey Critzer, Mar 29 2015
a(n) = A000203(n) - A000593(n). - Omar E. Pol, Apr 05 2016
L.g.f.: -log(Product_{ k>0 } (1-x^(2*k))) = Sum_{ n>=0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jul 04 2016
a(n) = A000203(n)*(1 - (1/A038712(n))). - Omar E. Pol, Aug 01 2018
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/24 = 0.411233... (A222171). - Amiram Eldar, Nov 06 2022
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000203(k) = 2 - A065442 = 0.393304... . - Amiram Eldar, Dec 14 2024

Corrected by Jaroslav Krizek, May 07 2011

A129194 a(n) = (n/2)^2*(3 - (-1)^n).

Original entry on oeis.org

0, 1, 2, 9, 8, 25, 18, 49, 32, 81, 50, 121, 72, 169, 98, 225, 128, 289, 162, 361, 200, 441, 242, 529, 288, 625, 338, 729, 392, 841, 450, 961, 512, 1089, 578, 1225, 648, 1369, 722, 1521, 800, 1681, 882, 1849, 968, 2025, 1058, 2209, 1152, 2401, 1250, 2601, 1352

Offset: 0

Paul Barry, Apr 02 2007

The numerator of the integral is 2,1,2,1,2,1,...; the moments of the integral are 2/(n+1)^2. See 2nd formula.
The sequence alternates between twice a square and an odd square, A001105(n) and A016754(n).
Partial sums of the positive elements give the absolute values of A122576. - Omar E. Pol, Aug 22 2011
Partial sums of the positive elements give A212760. - Omar E. Pol, Dec 28 2013
Conjecture: denominator of 4/n - 2/n^2. - Wesley Ivan Hurt, Jul 11 2016
Multiplicative because both A000290 and A040001 are. - Andrew Howroyd, Jul 25 2018

G. Pólya and G. Szegő, Problems and Theorems in Analysis II (Springer 1924, reprinted 1976), Part Eight, Chap. 1, Sect. 7, Problem 73.

Vincenzo Librandi, Table of n, a(n) for n = 0..960
Olivier Bordelles, A Multidimensional Cesaro Type Identity and Applications, J.
Int. Seq. 18 (2015) # 15.3.7.
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000290, A001105, A007434, A010713, A016742, A016754, A040001.

Cf. A061038, A061040, A061050, A105398, A129204, A152020, A222171, A309337.

[n^2*(3-(-1)^n)/4: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

A129194:=n->n^2*(3-(-1)^n)/4: seq(A129194(n), n=0..80); # Wesley Ivan Hurt, Jul 11 2016

Table[n^2*(3-(-1)^n)/4, {n,0,60}] (* Wesley Ivan Hurt, Jul 11 2016 *)
LinearRecurrence[{0,3,0,-3,0,1},{0,1,2,9,8,25},60] (* Harvey P. Dale, Dec 27 2023 *)

a(n) = lcm(2, n^2)/2; \\ Andrew Howroyd, Jul 25 2018

[n^2*(1+(n%2))/2 for n in range(61)] # G. C. Greubel, Apr 04 2023

G.f.: x*(1 + 2*x + 6*x^2 + 2*x^3 + x^4)/(1-x^2)^3.
a(n+1) = denominator((1/(2*Pi))*Integral_{t=0..2*Pi} exp(i*n*t)(-((Pi-t)/i)^2)), i=sqrt(-1).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n > 5. - Paul Curtz, Mar 07 2011
a(n) is the numerator of the coefficient of x^4 in the Maclaurin expansion of exp(-n*x^2). - Francesco Daddi, Aug 04 2011
O.g.f. as a Lambert series: x*Sum_{n >= 1} J_2(n)*x^n/(1 + x^n), where J_2(n) denotes the Jordan totient function A007434(n). See Pólya and Szegő. - Peter Bala, Dec 28 2013
From Ilya Gutkovskiy, Jul 11 2016: (Start)
E.g.f.: x*((2*x + 1)*sinh(x) + (x + 2)*cosh(x))/2.
Sum_{n>=1} 1/a(n) = 5*Pi^2/24. [corrected by Amiram Eldar, Sep 11 2022] (End)
a(n) = A000290(n) / A040001(n). - Andrew Howroyd, Jul 25 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/24 (A222171). - Amiram Eldar, Sep 11 2022
From Peter Bala, Jan 16 2024: (Start)
a(n) = Sum_{1 <= i, j <= n} (-1)^(1 + gcd(i,j,n)) = Sum_{d | n} (-1)^(d+1) * J_2(n/d), that is, the Dirichlet convolution of the pair of multiplicative functions f(n) = (-1)^(n+1) and the Jordan totient function J_2(n) = A007434(n). Hence this sequence is multiplicative. Cf. A193356 and A309337.
Dirichlet g.f.: (1 - 2/2^s)*zeta(s-2). (End)
a(n) = Sum_{1 <= i, j <= n} (-1)^(n + gcd(i, n)*gcd(j, n)) = Sum_{d|n, e|n} (-1)^(n+e*d) * phi(n/d)*phi(n/e). - Peter Bala, Jan 22 2024

More terms from Michel Marcus, Dec 28 2013

A245058 Decimal expansion of the real part of Li_2(I), negated.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Aug 21 2014

This is the decimal expansion of the real part of the dilogarithm of the square root of -1. The imaginary part is Catalan's number (A006752).

5*Pi^2/24 = 10 * (this constant) equals the asymptotic mean of the abundancy index of the even numbers. - Amiram Eldar, May 12 2023

			0.2056167583560283045590518958307531486523687376508498047169447786712509338004...

G. C. Greubel, Table of n, a(n) for n = 0..10000 (terms 0..104 from Robert G. Wilson v)
R. J. Mathar, Some definite integrals over a power multiplied by four modified Bessel functions vixra:1606.0141 (2016) eq. (39)
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (6.3.13)

Cf. A006752, A007949, A072691, A222171, A242301.

SetDefaultRealField(RealField(100)); R:=RealField(); Pi(R)^2/48; // G. C. Greubel, Aug 25 2018

RealDigits[ Re[ PolyLog[2, I]], 10, 111][[1]] (* or *) RealDigits[ Zeta[2]/8, 10, 111][[1]] (* or *) RealDigits[ Pi^2/48, 10, 111][[1]]

zeta(2)/8 \\ Charles R Greathouse IV, Aug 27 2014

(pi**2/48).n(200) # F. Chapoton, Mar 16 2020

Also equals -zeta(2)/8 = -Pi^2/48.
Also equals the Bessel moment Integral_{0..inf} x I_1(x) K_0(x)^2 K_1(x) dx. - Jean-François Alcover, Jun 05 2016
From Terry D. Grant, Sep 11 2016: (Start)
Equals Sum_{n>=0} (-1)^n/(2n+2)^2.
Equals (Sum_{n>=1} 1/(2n)^2)/2 = A222171/2. (End)
Equals Sum_{k>=1} A007949(k)/k^2. - Amiram Eldar, Jul 13 2020
Equals a tenth of integral_0^{pi/2} arccos[cos x/(1+2 cos x)]dx [Nahin]. - R. J. Mathar, May 22 2024
Equals Integral_{x>=0} x/(exp(2*x) + 1) dx. - Kritsada Moomuang, May 29 2025

A282097 Coefficients in q-expansion of (3E_2E_4 - 2*E_6 - E_2^3)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 12, 36, 112, 150, 432, 392, 960, 1053, 1800, 1452, 4032, 2366, 4704, 5400, 7936, 5202, 12636, 7220, 16800, 14112, 17424, 12696, 34560, 19375, 28392, 29160, 43904, 25230, 64800, 30752, 64512, 52272, 62424, 58800, 117936, 52022, 86640, 85176, 144000, 70602

Offset: 0

Seiichi Manyama, Feb 06 2017

Multiplicative because A000203 is. - Andrew Howroyd, Jul 25 2018

			a(6) = 1^3*6^2 + 2^3*3^2 + 3^3*2^2 + 6^3*1^2 = 432.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. this sequence (phi_{3, 2}), A282099 (phi_{5, 2}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282018 (E_2^3), A282019 (E_2*E_4).
Cf. A000203 (sigma(n)), A064987 (n*sigma(n)), this sequence (n^2*sigma(n)), A282211 (n^3*sigma(n)).
Cf. A222171.

[0] cat [n^2*DivisorSigma(1, n): n in [1..50]]; // Vincenzo Librandi, Mar 01 2018

a[0]=0;a[n_]:=(n^2)*DivisorSigma[1,n];Table[a[n],{n,0,41}] (* Indranil Ghosh, Feb 21 2017 *)
terms = 42; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[(3*Ei[2]*Ei[4] - 2*Ei[6] - Ei[2]^3)/1728 + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

a(n) = if (n==0, 0, n^2*sigma(n)); \\ Michel Marcus, Feb 21 2017

a(n) = (3*A282019(n) - 2*A013973(n) - A282018(n))/1728.
G.f.: phi_{3, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = n^2*A000203(n) for n > 0. - Seiichi Manyama, Feb 19 2017
G.f.: Sum_{k>=1} k^3*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, May 02 2018
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(e+1)-1)/(p-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-3).
Sum_{k=1..n} a(k) ~ (Pi^2/24) * n^4. (End)
From Peter Bala, Jan 22 2024: (Start)
a(n) = Sum_{1 <= i, j, k <= n} sigma_2( gcd(i, j, k, n) ).
a(n) = Sum_{1 <= i, j <= n} sigma_3( gcd(i, j, n) ).
a(n) = Sum_{d divides n} sigma_2(d) * J_3(n/d) = Sum_{d divides n} sigma_3(d) * J_2(n/d), where the Jordan totient functions J_2(n) = A007434(n) and J_3(n) = A059376(n). (End)

A275647 Decimal expansion of Pi^2/6 - Sum_{k>=1} 1/prime(k)^2.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Aug 04 2016

Decimal expansion of sum of squares of reciprocals of nonprime numbers.
Decimal expansion of the nonprime zeta function at 2.
Continued fraction [1; 5, 5, 3, 1, 2, 2, 6, 2, 2, 4, 1, 1, 93, 2, 1, 1, 5, 3, 5, 3, 2, 1, 2, 6, 1, 4, 5, 1, 34, 1, ...]
More generally, the nonprime zeta function at s equals Sum_{k>=1} (1/k^s - 1/prime(k)^s) = Product_{k>=1} 1/(1 - prime(k)^(-s)) - Sum_{k>=1} 1/prime(k)^s.
Floor(1/(zeta(s)-prime zeta(s)-1)) gives second term in continued fraction for nonprime zeta(s): 5, 36, 187, 852, 3663, 15280, 62692, 254760, 1029279, 4143617, ...
Dirichlet g.f. of A005171: nonprime zeta(s).

			1/1^2 + 1/4^2 + 1/6^2 + 1/8^2 + 1/9^2 + 1/10^2 + ... = 1.192686646807160937965871801813777255045718557966906015999...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Ilya Gutkovskiy, Nonprime zeta function.
Eric Weisstein's World of Mathematics, Riemann Zeta Function 2.
Eric Weisstein's World of Mathematics, Prime Zeta Function.

Cf. A002808, A008683, A013661, A018252, A062312, A085548, A111003, A222171.

RealDigits[Pi^2/6 - PrimeZetaP[2], 10, 120][[1]]
RealDigits[Zeta[2] - PrimeZetaP[2], 10, 120][[1]]

eps()=2.>>bitprecision(1.)
primezeta(s)=my(lm=s*log(2)); lm=lambertw(lm/eps())\lm; sum(k=1,lm, moebius(k)*log(abs(zeta(k*s)))/k)
zeta(2) - primezeta(2) \\ Charles R Greathouse IV, Aug 05 2016

Pi^2/6 - sumeulerrat(1/p, 2) \\ Amiram Eldar, Mar 19 2021

Equals zeta(2) - prime zeta(2) = A013661 - A085548.
Equals Sum_{k>=1} (1 - k*mu(k)*log(zeta(2*k)))/k^2, where mu(k) is the Moebius function (A008683).
Equals Sum_{k>=1} 1/A062312(k).
Equals Sum_{k>=1} 1/A018252(k)^2.
Equals 1 + Sum_{k>=1} 1/A002808(k)^2.
Equals A222171 + A111003 - A085548.

A284362 a(n) = Sum_{d|n, d = 0, 1, or 5 mod 6} d.

Original entry on oeis.org

1, 1, 1, 1, 6, 7, 8, 1, 1, 6, 12, 19, 14, 8, 6, 1, 18, 25, 20, 6, 8, 12, 24, 43, 31, 14, 1, 8, 30, 42, 32, 1, 12, 18, 48, 73, 38, 20, 14, 6, 42, 56, 44, 12, 6, 24, 48, 91, 57, 31, 18, 14, 54, 79, 72, 8, 20, 30, 60, 114, 62, 32, 8, 1, 84, 84, 68, 18, 24, 48, 72

Offset: 1

Seiichi Manyama, Mar 25 2017

			From _Peter Bala_, Dec 11 2020: (Start)
n = 20: n is not of the form m*(3*m +- 2), so e(n) = 0 and a(20) = a(19) + a(15) - a(12) - a(4) = 20 + 6 - 19 - 1 = 6;
n = 21: n = m*(3*m - 2) for m = 3, so e(n) = 21 and a(21) = 21 + a(20) + a(16) - a(13) - a(5) = 21 + 6 + 1 - 14 - 6 = 8;
n = 40: n = m*(3*m - 2) for m = 4, so e(n) = -40 and a(4) = -40 + a(39) + a(35) - a(32) - a(24) + a(19) + a(7) = -40 + 14 + 48  - 1 - 43 + 20 + 8 = 6. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A089802 (f(-x, -x^5)), A195848 (1/f(-x, -x^5)), A222171.

Cf. Sum_{d|n, d = 0, 1, or k-1 mod k} d: A000203 (k=3), A284361 (k=5), this sequence (k=6), A284363 (k=7), A284372 (k=12).

Table[Sum[If[Mod[d, 6] <2 || Mod[d, 6]==5, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 25 2017 *)

a(n) = sumdiv(n, d, ((d + 1) % 6 < 3) * d); \\ Amiram Eldar, Apr 12 2024

From Peter Bala, Dec 11 2020: (Start)
O.g.f.: Sum_{k >= 1} ( (6*k)*x^(6*k)/(1 - x^(6*k)) + (6*k-1)*x^(6*k-1)/(1 - x^(6*k-1)) + (6*k-5)*x^(6*k-5)/(1 - x^(6*k-5)) ).
Define a(n) = 0 for n < 1. Then a(n) = e(n) + a(n-1) + a(n-5) - a(n-8) - a(n-16) + + - -, where [1, 5, 8, 16, ...] is the sequence of generalized octagonal numbers A001082, and e(n) = (-1)^(m+1)*n if n is a generalized octagonal number of the form m*(3*m+-2); otherwise e(n) = 0. Examples of this recurrence are given below. (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/24 = A222171 = 0.411233... . - Amiram Eldar, Apr 12 2024

A111932 Expansion of q * (psi(q) * psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 1, 4, 6, 2, 8, 8, 1, 12, 12, 4, 14, 16, 6, 16, 18, 2, 20, 24, 8, 24, 24, 8, 31, 28, 1, 32, 30, 12, 32, 32, 12, 36, 48, 4, 38, 40, 14, 48, 42, 16, 44, 48, 6, 48, 48, 16, 57, 62, 18, 56, 54, 2, 72, 64, 20, 60, 60, 24, 62, 64, 8, 64, 84, 24, 68, 72, 24, 96, 72, 8, 74, 76, 31

Offset: 1

Michael Somos, Aug 21 2005, Apr 18 2007

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

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

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 223 Entry 3(iii).
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 87, Eq. (33.2).

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033762, A131946, A222171.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

a[ n_] := If[ n < 1, 0, Sum[ Mod[n/d, 2] d KroneckerSymbol[ 9, d], { d, Divisors[ n]}]]; (* Michael Somos, Sep 19 2013 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^2] QPochhammer[ q^6])^4 / (QPochhammer[ q] QPochhammer[ q^3])^2, {q, 0, n}]; (* Michael Somos, Sep 19 2013 *)

{a(n) = if( n<1, 0, sumdiv(n, d, (n/d % 2) * d * (d%3>0)))};

{a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p==2, p^e, if( p==3, 1, (p^(e+1) - 1) / (p-1)))))) };

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^4 / (eta(x + A) * eta(x^3 + A))^2, n))};

A = ModularForms( Gamma0(6), 2, prec=50) . basis();  A[1] + 2*A[2]; # Michael Somos, Sep 19 2013

Expansion of (1/3) * (b(q^2)^2 / b(q))* (c(q^2)^2 / c(q)) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of (eta(q^2) * eta(q^6))^4 / (eta(q) * eta(q^3))^2 in powers of q.
Euler transform of period 6 sequence [ 2, -2, 4, -2, 2, -4, ...].
Multiplicative with a(2^e) = 2^e, a(3^e) = 1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u*w * (u - 4*v) - v * (v - 4*w)^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (3/4) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A131946. - Michael Somos, Sep 19 2013
G.f.: Sum_{k>0} k * x^k * (1 - x^(2*k))^2 / (1 - x^(6*k)) = x * Product_{k>0} ((1 + x^k) * (1 + x^(3*k)))^4 * ((1 - x^k) * (1 - x^(3*k)))^2.
a(3*n) = a(n), a(2*n) = 2 * a(n).
Convolution square of A033762. - Michael Somos, Sep 19 2013
From Amiram Eldar, Sep 12 2023: (Start)
Dirichlet g.f.: (1 - 1/2^s) * (1 - 1/3^(s-1)) * zeta(s-1) * zeta(s).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/24 = 0.411233... (A222171). (End)

A271342 Sum of all even divisors of all positive integers <= n.

Original entry on oeis.org

0, 2, 2, 8, 8, 16, 16, 30, 30, 42, 42, 66, 66, 82, 82, 112, 112, 138, 138, 174, 174, 198, 198, 254, 254, 282, 282, 330, 330, 378, 378, 440, 440, 476, 476, 554, 554, 594, 594, 678, 678, 742, 742, 814, 814, 862, 862, 982, 982, 1044, 1044, 1128, 1128, 1208, 1208, 1320, 1320, 1380, 1380, 1524, 1524, 1588, 1588, 1714, 1714

Offset: 1

Omar E. Pol, Apr 08 2016

a(n) is also the sum of all even divisors of all even positive integers <= n.
a(n) is also the total number of parts in all partitions of all positive integers <= n into an even number of equal parts. - Omar E. Pol, Jun 04 2017
The bisection of this sequence equals twice A024916 (see formulas). - Michel Marcus, Dec 14 2017

			For n = 6 the divisors of all positive integers <= 6 are [1], [1, 2], [1, 3], [1, 2, 4], [1, 5], [1, 2, 3, 6] and the even divisors of all positive integers <= 6 are [2], [2, 4], [2, 6], so a(6) = 2 + 2 + 4 + 2 + 6 = 16. On the other hand the sum of all the divisors of all positive integers <= 6/2 are [1] + [1 + 2] + [1 + 3] = A024916(3) = 8, so a(6) = 2*8 = 16.
For n = 10, (floor(10/2) = 5) numbers have divisor 2, (floor(10/4) = 2) numbers have divisor 4, ..., (floor(10/10) = 1) numbers have divisor 10. Therefore, a(10) = 5 * 2 + 2 * 4 + 1 * 6 + 1 * 8 + 1 * 10 = 42. - _David A. Corneth_, Jun 06 2017

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000203, A006128, A024916, A078471, A146076, A222171, A271343.

Partial sums of A146076.

Accumulate@ Array[DivisorSum[#, # &, EvenQ] &, 65] (* Michael De Vlieger, Jun 06 2017 *)

a(n) = sum(k=1, n, sumdiv(k, d, (1-d%2)*d)); \\ Michel Marcus, Jun 05 2017

a(n) = 2 * sum(k=1, n\2, k*(n\(k<<1))) \\ David A. Corneth, Jun 06 2017

def A271342(n): return sum(k*((n>>1)//k) for k in range(1, (n>>1)+1))<<1 # Chai Wah Wu, Apr 26 2023

from math import isqrt
def A271342(n): return -(s:=isqrt(m:=n>>1))**2*(s+1) + sum((q:=m//k)*((k<<1)+q+1) for k in range(1,s+1)) # Chai Wah Wu, Oct 21 2023

a(1) = 0.
a(n) = 2*A024916((n-1)/2), if n is odd and n > 1.
a(n) = 2*A024916(n/2), if n is even.
a(n) = A024916(n) - A078471(n).
For n > 1, a(2*n + 1) = a(2*n). - David A. Corneth, Jun 06 2017
a(n) = c * n^2 + O(n*log(n)), where c = Pi^2/24 = 0.411233... (A222171). - Amiram Eldar, Nov 27 2023

cp's OEIS Frontend

A005843 The nonnegative even numbers: a(n) = 2n.

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A001105 a(n) = 2*n^2.

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A146076 Sum of even divisors of n.

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A129194 a(n) = (n/2)^2*(3 - (-1)^n).

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A245058 Decimal expansion of the real part of Li_2(I), negated.

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A282097 Coefficients in q-expansion of (3E_2E_4 - 2*E_6 - E_2^3)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.