A024916 -id:A024916 - cp's OEIS frontend

A056550 Numbers k such that Sum_{j=1..k} sigma(j) is divisible by k, where sigma(j) = sum of divisors of j (A000203).

1, 2, 8, 11, 17, 63, 180, 259, 818, 2161, 4441, 8305, 11998, 694218, 3447076, 4393603, 57402883, 73459800, 121475393, 2068420025, 5577330586, 13320495021, 35297649260, 138630178659, 988671518737, 1424539472772, 3028785109162, 13702718147734, 21320824383487

Offset: 1

Asher Auel, Jun 06 2000

			a(3) = 8 is in the sequence because A024916(8) / 8 = 56 / 8 = 7 is an integer. [_Jaroslav Krizek_, Dec 07 2009]

Cf. A000203, A024916, A168133, A168132.

f := []: for i from 1 to 9000 do if add(sigma(n), n=1..i) mod i = 0 then f := [op(f),i] fi; od; f;

k=10^4;a[1]=1;a[n_]:=a[n]=DivisorSigma[1,n]+a[n-1]; s=a/@Range@k;Select[Range@k,Divisible[s[[#]],#]&] (* Ivan N. Ianakiev, Apr 30 2016 *)
Module[{nn=44*10^5,ds},ds=Accumulate[DivisorSigma[1,Range[nn]]];Select[ Thread[{ds,Range[nn]}],Divisible[#[[1]],#[[2]]]&]][[All,2]] (* The program generates the first 16 terms of the sequence. To generate more, increase the value of nn. *) (* Harvey P. Dale, Dec 04 2018 *)

is(n)=sum(k=1,n,n\k*k)%n==0 \\ Charles R Greathouse IV, Feb 14 2013

Values of k for which A024916(k)/k is an integer.

More terms from Jud McCranie, Jul 04 2000
a(19)-a(24) from Donovan Johnson, Dec 29 2008
a(25) from Donovan Johnson, Jun 16 2011
a(26) from Jud McCranie, Dec 17 2024
a(27) from Jud McCranie, Dec 22 2024
a(28) from Jud McCranie, Apr 03 2025
a(29) from Jud McCranie, May 04 2025

A046895 Sizes of successive clusters in Z^4 lattice.

Original entry on oeis.org

1, 9, 33, 65, 89, 137, 233, 297, 321, 425, 569, 665, 761, 873, 1065, 1257, 1281, 1425, 1737, 1897, 2041, 2297, 2585, 2777, 2873, 3121, 3457, 3777, 3969, 4209, 4785, 5041, 5065, 5449, 5881, 6265, 6577, 6881, 7361, 7809, 7953, 8289, 9057

Offset: 0

N. J. A. Sloane

Number of lattice points inside or on the 4-sphere x^2 + y^2 + z^2 + u^2 = n. - T. D. Noe, Mar 14 2009

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 (terms up to 1000 from Noe)
A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Volume 44, Issue 12, page 607, 1964.

Partial sums of A000118.

Cf. A117609

Accumulate[ Table[ SquaresR[4, n], {n, 0, 42}]] (* Jean-François Alcover, May 11 2012 *)
QP = QPochhammer; s = (QP[q^2]^5/(QP[q]^2*QP[q^4]^2))^4/(1-q) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015, after Joerg Arndt *)

q='q+O('q^66);
Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^4/(1-q))
/* Joerg Arndt, Apr 08 2013 */

from math import isqrt
def A046895(n): return 1+((-(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))&-1)<<2)+(((t:=isqrt(m:=n>>2))**2*(t+1)-sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))&-1)<<4) # Chai Wah Wu, Jun 21 2024

a(n) = A122510(4,n). a(n^2) = A055410(n). - R. J. Mathar, Apr 21 2010
G.f.: T3(q)^4/(1-q) where T3(q) = 1 + 2*Sum_{k>=1} q^(k^2). - Joerg Arndt, Apr 08 2013
Pi^2/2 * (sqrt(n)-1)^4 < a(n) < Pi^2/2 * (sqrt(n)+1)^4 for n > 0. - Charles R Greathouse IV, Feb 17 2015
a(n) = Pi^2/2 * n^2 + O(n (log n)^(2/3)) using a result of Walfisz. - Charles R Greathouse IV, Feb 18 2015
a(n) = 1 + 8*A024916(n) - 32*A024916(floor(n/4)) by Jacobi's four-square theorem. - Peter J. Taylor, Jun 03 2020

A078471 Sum of all odd divisors of all positive integers <= n.

Original entry on oeis.org

1, 2, 6, 7, 13, 17, 25, 26, 39, 45, 57, 61, 75, 83, 107, 108, 126, 139, 159, 165, 197, 209, 233, 237, 268, 282, 322, 330, 360, 384, 416, 417, 465, 483, 531, 544, 582, 602, 658, 664, 706, 738, 782, 794, 872, 896, 944, 948, 1005, 1036, 1108, 1122, 1176, 1216

Offset: 1

Benoit Cloitre, Dec 31 2002

nonn

The subsequence of primes begins: 2, 7, 13, 17, 61, 83, 107, 139, 197, 233, then no more through a(54). [Jonathan Vos Post, Feb 14 2010]

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

Alois P. Heinz, Table of n, a(n) for n = 1..10000
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, arXiv:math/9904028 [math.MG], 1999.
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276.

Partial sums of A000593.

Cf. A024916, A271342, A000203, A326124.

[&+[&+[d:d in Divisors(k)|IsOdd(d)]:k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Aug 28 2019

with(numtheory):
b:= n-> add(d, d=select(x-> x::odd, divisors(n))):
a:= proc(n) option remember; b(n)+`if`(n=1, 0, a(n-1)) end:
seq(a(n), n=1..60);  # Alois P. Heinz, Sep 25 2015

a[n_] := Sum[DivisorSum[k, (-1)^(# + 1) k/# &], {k, 1, n}]; Array[a, 60] (* Jean-François Alcover, Dec 07 2015 *)
Accumulate[Table[Total[Select[Divisors[n],OddQ]],{n,60}]] (* Harvey P. Dale, Sep 15 2024 *)

a(n)=sum(v=1,n,sumdiv(v,d,(-1)^(d+1)*v/d))

a(n) = sum(k=1, n, sumdiv(k, d, (d%2)*d)); \\ Michel Marcus, Apr 09 2016

def A078471(n): return sum(k*(n//k) for k in range((n>>1)+1, n+1)) + sum(k*(n//k-((n>>1)//k<<1)) for k in range(1, (n>>1)+1)) # Chai Wah Wu, Apr 26 2023

from math import isqrt
def A078471(n): return (t:=isqrt(m:=n>>1))**2*(t+1) - sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))-((s:=isqrt(n))**2*(s+1) - sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{k=1..n} A000593(k).
a(n) is asymptotic to c*n^2 where c = Pi^2/24.
a(n) = A024916(n) - A271342(n). - Omar E. Pol, Apr 08 2016
G.f.: (1/(1 - x))*Sum_{k>=1} k*x^k/(1 + x^k). - Ilya Gutkovskiy, Dec 23 2016
From Ridouane Oudra, Aug 28 2019: (Start)
a(n) = Sum_{k=1..n} (sigma(2k) - 2*sigma(k)), where sigma = A000203.
a(n) = A326124(n) - 2*A024916(n). (End)

Better definition from Omar E. Pol, Apr 09 2016

A259179 Number of Dyck paths described in A237593 that contain the point (n,n) in the diagram of the symmetric representation of sigma.

Original entry on oeis.org

1, 2, 2, 0, 2, 1, 3, 0, 3, 0, 1, 2, 2, 0, 4, 0, 1, 3, 0, 2, 0, 2, 3, 0, 1, 4, 0, 2, 0, 3, 0, 3, 0, 1, 1, 4, 0, 2, 0, 4, 0, 3, 0, 1, 2, 0, 4, 0, 2, 0, 0, 5, 0, 3, 0, 1, 3, 0, 4, 0, 2, 0, 1, 0, 5, 0, 2, 1, 0, 1, 4, 0, 4, 0, 2, 0, 2, 0, 5, 0, 3, 0, 0, 0, 1, 5, 0, 2, 2, 0, 2, 0, 3, 0, 5, 0, 3, 0, 1, 0, 0, 6

Offset: 1

Omar E. Pol, Aug 11 2015

nonn

Since the diagram of the symmetric representation of sigma is also the top view of the stepped pyramid described in A245092, and the diagram is also the top view of the staircase described in A244580, so we have that a(n) is also the height difference (or length of the vertical line segment) at the point (n,n) in the main diagonal of the mentioned structures.
a(n) is the number of occurrences of n in A240542. - Omar E. Pol, Dec 09 2016
Nonzero terms give A280919, the first differences of A071562. - Omar E. Pol, Apr 17 2018
Also first differences of A244367. Where records occur gives A279286. - Omar E. Pol, Apr 20 2020

			Illustration of initial terms:
--------------------------------------------------------
                           Diagram with 15 Dyck paths
n   A000203(n)  a(n)         to evaluate a(1)..a(10)
--------------------------------------------------------
.                         _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1        1        1      |_| | | | | | | | | | | | | | |
2        3        2      |_ _|_| | | | | | | | | | | | |
3        4        2      |_ _|  _|_| | | | | | | | | | |
4        7        0      |_ _ _|    _|_| | | | | | | | |
5        6        2      |_ _ _|  _|  _ _|_| | | | | | |
6       12        1      |_ _ _ _|  _| |  _ _|_| | | | |
7        8        3      |_ _ _ _| |_ _|_|    _ _|_| | |
8       15        0      |_ _ _ _ _|  _|     |  _ _ _|_|
9       13        3      |_ _ _ _ _| |      _|_| |
10      18        0      |_ _ _ _ _ _|  _ _|    _|
.                        |_ _ _ _ _ _| |  _|  _|
.                        |_ _ _ _ _ _ _| |_ _|
.                        |_ _ _ _ _ _ _| |
.                        |_ _ _ _ _ _ _ _|
.                        |_ _ _ _ _ _ _ _|
.
For n = 3 there are two Dyck paths that contain the point (3,3) so a(3) = 2.
For n = 4 there are no Dyck paths that contain the point (4,4) so a(4) = 0.

Cf. A000203, A024916, A071562, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A240542, A244050, A244367, A244580, A245092, A249351, A256533, A259179, A262626, A279286, A280919.

a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
a259179[n_] := Module[{t=Table[0, n], k=1, d=1}, While[d<=n, t[[d]]+=1; d=a240542[++k]]; t] (* a(1..n) *)
a259179[102] (* Hartmut F. W. Hoft, Aug 06 2020 *)

More terms from Omar E. Pol, Dec 09 2016

A195055 Decimal expansion of Pi^2/3.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 04 2011

			3.289868133696452872944830333292050378438...

Marc Briane and Gilles Pagès, Théorie de l'Intégration, Vuibert, 2004, 3ème édition, exercice 12.15, p. 256.

George E. Andrews, Partitions with short sequences and mock theta functions, Proceedings of the National Academy of Sciences, Vol. 102, No. 13 (2005), pp. 4666-4671.
Index entries for transcendental numbers

Cf. A002388, A102753, A091476, 2*A013661, A164102.
Cf. A024916 (partial sums of A000203).
Cf. A001008/A002805, A145426.

pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*pi^2/3))); // Vincenzo Librandi, Jan 12 2016

RealDigits[Pi^2/3, 10, 105][[1]] (* T. D. Noe, Oct 05 2011 *)

2*zeta(2) \\ Charles R Greathouse IV, Jan 20 2022

sumnumrat(1/x^2,-oo) \\ Charles R Greathouse IV, Jan 20 2022

Equals 3 + A145426.
Equals -Sum_{n>=1} Psi_2(n), where Psi_2 is the tetragamma function. - Istvan Mezo, Oct 25 2012
Equals Integral_{x=0..1} (log(x)/(x - 1))^2 dx. - Jean-François Alcover, Mar 21 2013
Equals Integral_{x=-oo..oo} x^2/sinh(x)^2 dx. - Amiram Eldar, Aug 06 2020
Equals Integral_{x=0..oo} (log(x+1)/x)^2 dx (reference Briane and Pagès). - Bernard Schott, Feb 13 2022
Equals Sum_{n>=1} H(n) * binomial(2*n, n) / (n * 4^n), where H(n) is the n-th harmonic number. - Antonio Graciá Llorente, Apr 04 2025

Extended by T. D. Noe, Oct 05 2011

A244049 Sum of all proper divisors of all positive integers <= n.

Original entry on oeis.org

0, 0, 0, 2, 2, 7, 7, 13, 16, 23, 23, 38, 38, 47, 55, 69, 69, 89, 89, 110, 120, 133, 133, 168, 173, 188, 200, 227, 227, 268, 268, 298, 312, 331, 343, 397, 397, 418, 434, 483, 483, 536, 536, 575, 607, 632, 632, 707, 714, 756, 776, 821, 821, 886, 902

Offset: 1

Omar E. Pol, Jun 24 2014

The proper divisors of n are all divisors except 1 and n itself. Therefore noncomposite numbers have no proper divisors.
For the sum of all aliquot divisors of all positive integers <= n see A153485.
For the sum all divisors of all positive integers <= n see A024916.
a(n) = a(n - 1) if and only if n is prime.
For n >= 3 a(n) equals the area of an arrowhead-shaped polygon formed by two zig-zag paths and the Dyck path described in the n-th row of A237593 as shown in the Links section. Note that there is a similar diagram of A153485(n) in A153485. - Omar E. Pol, Jun 14 2022

			a(4) = 2 because the only proper divisor of 4 is 2 and the previous n contributed no proper divisors to the sum.
a(5) = 2 because 5 is prime and contributes no proper divisors to the sum.
a(6) = 7 because the proper divisors of 6 are 2 and 3, which add up to 5, and a(5) + 5 = 2 + 5 = 7.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Omar E. Pol, Illustration of initial terms with arrowhead-shaped polygons.

Partial sums of A048050.

Cf. A000040, A000203, A000290, A001065, A004125, A008578, A024916, A027750, A034856, A153485, A161680, A237591, A237593, A245092, A262626.

propDivsRunSum[1] := 0; propDivsRunSum[n_] := propDivsRunSum[n] = propDivsRunSum[n - 1] + (Plus@@Divisors[n]) - (n + 1); Table[propDivsRunSum[n], {n, 60}] (* Alonso del Arte, Jun 30 2014 *)
Accumulate[Join[{0},Table[Total[Most[Divisors[n]]]-1,{n,2,60}]]] (* Harvey P. Dale, Aug 12 2016 *)
Accumulate[Join[{0}, Table[DivisorSigma[1, n] - n - 1, {n, 2, 55}]]] (* Amiram Eldar, Jun 18 2022 *)

a(n) = sum(k=2, n, sigma(k)-k-1); \\ Michel Marcus, Mar 30 2021

from math import isqrt
def A244049(n): return ((-n*(n+3)-(s:=isqrt(n))**2*(s+1) + sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1)))>>1)+1 # Chai Wah Wu, Oct 21 2023

a(n) = A024916(n) - A034856(n).
a(n) = A153485(n) - n + 1.
G.f.: (1/(1 - x))*Sum_{k>=2} k*x^(2*k)/(1 - x^k). - Ilya Gutkovskiy, Jan 22 2017
a(n) = A161680(n-1) - A004125(n). - Omar E. Pol, Mar 25 2021
a(n) = A000290(n) - A034856(n) - A004125(n). - Omar E. Pol, Mar 26 2021
a(n) = c * n^2 + O(n*log(n)), where c = Pi^2/12 - 1/2 = 0.322467... . - Amiram Eldar, Nov 27 2023

A054541 Sum of first n terms equals n-th prime.

Original entry on oeis.org

2, 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6

Offset: 1

G. L. Honaker, Jr., Apr 09 2000

nonn

Except for first term, same as A001223.
First differences of A182986. - Omar E. Pol, Oct 31 2013
A075526 is 1 together with A001223. This is 2 together with A001223. A125266 is 3 together with A001223. - Omar E. Pol, Nov 01 2013
Convolved with A024916 gives A086718. - Omar E. Pol, Dec 23 2021

Partial sums give A000040.

Cf. A001223, A075526, A086718, A024916, A125266, A182986.

Join[{2},Differences[Prime[Range[100]]]] (* Paolo Xausa, Oct 25 2023 *)

a(n) = if (n==1, 2, prime(n) - prime(n-1)); \\ Michel Marcus, Oct 31 2013

More terms from James Sellers, Apr 11 2000

A340793 Sequence whose partial sums give A000203.

Original entry on oeis.org

1, 2, 1, 3, -1, 6, -4, 7, -2, 5, -6, 16, -14, 10, 0, 7, -13, 21, -19, 22, -10, 4, -12, 36, -29, 11, -2, 16, -26, 42, -40, 31, -15, 6, -6, 43, -53, 22, -4, 34, -48, 54, -52, 40, -6, -6, -24, 76, -67, 36, -21, 26, -44, 66, -48, 48, -40, 10, -30, 108, -106, 34, 8

Offset: 1

Omar E. Pol, Jan 21 2021

Essentially a duplicate of A053222.
Convolved with the nonzero terms of A000217 gives A175254, the volume of the stepped pyramid described in A245092.
Convolved with the nonzero terms of A046092 gives A244050, the volume of the stepped pyramid described in A244050.
Convolved with A000027 gives A024916.
Convolved with A000041 gives A138879.
Convolved with A000070 gives the nonzero terms of A066186.
Convolved with the nonzero terms of A002088 gives A086733.
Convolved with A014153 gives A182738.
Convolved with A024916 gives A000385.
Convolved with A036469 gives the nonzero terms of A277029.
Convolved with A091360 gives A276432.
Convolved with A143128 gives the nonzero terms of A000441.
For the correspondence between divisors and partitions see A336811.

1 together with A053222.
Cf. A000203 (partial sums).
Cf. A000027, A000041, A000070, A000217, A000385, A000441, A002088, A002961, A014153, A024916, A036469, A046092, A053223, A053224, A053225, A053226, A053227, A066186, A086733, A091360, A138879, A143128, A175254, A182738, A237593, A244050, A245092, A276432, A277029, A336811.

a:= n-> (s-> s(n)-s(n-1))(numtheory[sigma]):
seq(a(n), n=1..77);  # Alois P. Heinz, Jan 21 2021

Join[{1}, Differences @ Table[DivisorSigma[1, n], {n, 1, 100}]] (* Amiram Eldar, Jan 21 2021 *)

a(n) = if (n==1, 1, sigma(n)-sigma(n-1)); \\ Michel Marcus, Jan 22 2021

a(n) = A053222(n-1) for n>1. - Michel Marcus, Jan 22 2021

A024919 a(n) = Sum_{k=1..n} (-1)^kkfloor(n/k).

Original entry on oeis.org

-1, 0, -4, 1, -5, -1, -9, 4, -9, -3, -15, 5, -9, -1, -25, 4, -14, -1, -21, 9, -23, -11, -35, 17, -14, 0, -40, 0, -30, -6, -38, 23, -25, -7, -55, 10, -28, -8, -64, 14, -28, 4, -40, 20, -58, -34, -82, 34, -23, 8, -64, 6, -48, -8, -80, 24, -56, -26, -86, 34, -28, 4, -100, 25, -59

Offset: 1

Clark Kimberling

sign

n - 2*[ n/2 ] + 3*[ n/3 ] - ... + m*n*[ n/n ], where m = (-1)^(n+1).

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

The zeros are A072663.
Partial sums of A002129.
Cf. A024916, A309124.

[&+[(-1)^k*k*(n div k): k in [1..n]]: n in [1..70]]; // Vincenzo Librandi, Jul 28 2019

f[n_] := Sum[(-1)^i*i*Floor[n/i], {i, 1, n}]; Table[ f[n], {n, 1, 85}]

a(n) = sum(k=1, n, (-1)^k*k*floor(n/k));

from math import isqrt
def A024919(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))<<1)+((s:=isqrt(n))**2*(s+1)-sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) # Chai Wah Wu, Oct 22 2023

a(n) = 4*A024916(floor(n/2)) - A024916(n). - Vladeta Jovovic, Oct 15 2002
G.f.: 1/(1-x) * Sum_{n>=1} n*x^n*(3*x^n-1)/(1-x^(2*n)). - Vladeta Jovovic, Oct 15 2002
G.f.: -1/(1-x) * Sum_{k>=1} x^k/(1+x^k)^2 = 1/(1-x) * Sum_{k>=1} k * (-x)^k/(1-x^k). - Seiichi Manyama, Oct 29 2023

Edited by Robert G. Wilson v, Aug 17 2002

A182738 Partial sums of A066186.

Original entry on oeis.org

1, 5, 14, 34, 69, 135, 240, 416, 686, 1106, 1722, 2646, 3959, 5849, 8489, 12185, 17234, 24164, 33474, 46014, 62646, 84690, 113555, 151355, 200305, 263641, 344911, 449015, 581400, 749520, 961622, 1228790, 1563509, 1982049

Offset: 1

Omar E. Pol, Jan 22 2011

a(n) is also the volume of a three-dimensional version of the section model of partitions: the 3D illustrations in A135010 show boxes with face areas of 1 X 1, 2 X 2, 3 X 3, 4 X 5, 5 X 7 units along the m and p(m) axis, which is sequence A066186. Assuming that the boxes are 1 unit deep, the total volume of all boxes up to layer n is a(n). See the first two links.
From Omar E. Pol, Jan 20 2021: (Start)
a(n) is the sum of all parts of all partitions of all positive integers <= n.
Convolution of A000203 and A000070.
Convolution of A024916 and A000041.
Convolution of A175254 and A002865.
Convolution of A340793 and A014153.
Row sums of triangles A340527, A340531, A340579.
Consider a symmetric tower (a polycube) in which the terraces are the symmetric representation of sigma (n..1) respectively starting from the base (cf. A237270, A237593). The total area of the terraces equals A024916(n), the same as the area of the base.
The levels of the terraces starting from the base are the first n terms of A000070, that is A000070(0)..A000070(n-1), hence the differences between two successive levels give the partition numbers A000041, that is A000041(0)..A000041(n-1).
a(n) is the volume (or the total number of unit cubes) of the polycube.
That is due to the correspondence between divisors and partitions (cf. A336811).
The symmetric tower is a member of the family of the pyramid described in A245092.
The growth of the volume of the polycube represents every convolution mentioned above. (End)

			a(6) = 135 because the volume V(6) = p(1) + 2*p(2) + 3*p(3) + 4*p(4) + 5*p(5) + 6*p(6) = 1 + 2*2 + 3*3 + 4*5 + 5*7 + 6*11 = 1 + 4 + 9 + 20 + 35 + 66 = 135 where p(n) = A000041(n).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Omar E. Pol, Illustration of a(6) = 135, the polycube contains 135 unit cubes.
Omar E. Pol, Illustration of a(9) = 686, the polycube contains 686 unit cubes.

Cf. A000041, A000070, A000203, A002865, A014153, A024916, A066186, A135010, A175254, A237270, A237593, A245092, A336811, A340527, A340531, A340579, A340793.

With[{no=35},Accumulate[PartitionsP[Range[no]]Range[no]]] (* Harvey P. Dale, Feb 02 2011 *)

a(n) = n*A000070(n) - A014153(n-1). - Vaclav Kotesovec, Jun 23 2015
a(n) ~ sqrt(n) * exp(Pi*sqrt(2*n/3)) / (Pi*2^(3/2)) * (1 + (11*Pi/(24*sqrt(6)) - sqrt(6)/Pi)/sqrt(n) + (73*Pi^2/6912 - 3/16)/n). - Vaclav Kotesovec, Jun 23 2015, extended Nov 04 2016
G.f.: x*f'(x)/(1 - x), where f(x) = Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 10 2017

cp's OEIS Frontend

A056550 Numbers k such that Sum_{j=1..k} sigma(j) is divisible by k, where sigma(j) = sum of divisors of j (A000203).

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A046895 Sizes of successive clusters in Z^4 lattice.

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A078471 Sum of all odd divisors of all positive integers <= n.

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A259179 Number of Dyck paths described in A237593 that contain the point (n,n) in the diagram of the symmetric representation of sigma.

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A195055 Decimal expansion of Pi^2/3.

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A244049 Sum of all proper divisors of all positive integers <= n.

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A024919 a(n) = Sum_{k=1..n} (-1)^kkfloor(n/k).