A072692 -id:A072692 - cp's OEIS frontend

A024916 a(n) = Sum_{k=1..n} k*floor(n/k); also Sum_{k=1..n} sigma(k) where sigma(n) = sum of divisors of n (A000203).

1, 4, 8, 15, 21, 33, 41, 56, 69, 87, 99, 127, 141, 165, 189, 220, 238, 277, 297, 339, 371, 407, 431, 491, 522, 564, 604, 660, 690, 762, 794, 857, 905, 959, 1007, 1098, 1136, 1196, 1252, 1342, 1384, 1480, 1524, 1608, 1686, 1758, 1806, 1930, 1987, 2080, 2152

Offset: 1

Clark Kimberling

Row sums of triangle A128489. E.g., a(5) = 15 = (10 + 3 + 1 + 1), sum of row 4 terms of triangle A128489. - Gary W. Adamson, Jun 03 2007
Row sums of triangle A134867. - Gary W. Adamson, Nov 14 2007
a(10^4) = 82256014, a(10^5) = 8224740835, a(10^6) = 822468118437, a(10^7) = 82246711794796; see A072692. - M. F. Hasler, Nov 22 2007
Equals row sums of triangle A158905. - Gary W. Adamson, Mar 29 2009
n is prime if and only if a(n) - a(n-1) - 1 = n. - Omar E. Pol, Dec 31 2012
Also the alternating row sums of A236104. - Omar E. Pol, Jul 21 2014
a(n) is also the total number of parts in all partitions of the positive integers <= n into equal parts. - Omar E. Pol, Apr 30 2017
a(n) is also the total area of the terraces of the stepped pyramid with n levels described in A245092. - Omar E. Pol, Nov 04 2017
a(n) is also the area under the Dyck path described in the n-th row of A237593 (see example). - Omar E. Pol, Sep 17 2018
From Omar E. Pol, Feb 17 2020: (Start)
Convolution of A340793 and A000027.
Convolved with A340793 gives A000385. (End)
a(n) is also the number of cubic cells (or cubes) in the n-th level starting from the top of the stepped pyramid described in A245092. - Omar E. Pol, Jan 12 2022

			From _Omar E. Pol_, Aug 20 2021: (Start)
For n = 6 the sum of all divisors of the first six positive integers is [1] + [1 + 2] + [1 + 3] + [1 + 2 + 4] + [1 + 5] + [1 + 2 + 3 + 6] = 1 + 3 + 4 + 7 + 6 + 12 = 33, so a(6) = 33.
On the other hand the area under the Dyck path of the 6th diagram as shown below is equal to 33, so a(6) = 33.
Illustration of initial terms:                        _ _ _ _
                                        _ _ _        |       |_
                            _ _ _      |     |       |         |_
                  _ _      |     |_    |     |_ _    |           |
          _ _    |   |_    |       |   |         |   |           |
    _    |   |   |     |   |       |   |         |   |           |
   |_|   |_ _|   |_ _ _|   |_ _ _ _|   |_ _ _ _ _|   |_ _ _ _ _ _|
.
    1      4        8          15           21             33         (End)

Hardy and Wright, "An introduction to the theory of numbers", Oxford University Press, fifth edition, p. 266.

Daniel Mondot, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Vaclav Kotesovec, Plot of (a(n) - Pi^2*n^2/12) / (n*log(n)^(2/3)) for n = 2..100000.
P. L. Patodia (pannalal(AT)usa.net), PARI program for A072692 and A024916.
Peter Polm, C# program for A024916.
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 A000203.
Cf. A056550, A104471(2*n-1, n), A123229, A128489, A000217, A134867, A072692, A158905, A237593, A245092, A006218, A222548, A092406, A160664.
Cf. A000385, A010766, A340793.

a024916 n = sum $ map (\k -> k * div n k) [1..n]
-- Reinhard Zumkeller, Apr 20 2015

[(&+[DivisorSigma(1, k): k in [1..n]]): n in [1..60]]; // G. C. Greubel, Mar 15 2019

A024916 := proc(n)
    add(numtheory[sigma](k),k=0..n) ;
end proc: # Zerinvary Lajos, Jan 11 2009
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 0,
      numtheory[sigma](n)+a(n-1))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 12 2019

Table[Plus @@ Flatten[Divisors[Range[n]]], {n, 50}] (* Alonso del Arte, Mar 06 2006 *)
Table[Sum[n - Mod[n, m], {m, n}], {n, 50}] (* Roger L. Bagula and Gary W. Adamson, Oct 06 2006 *)
a[n_] := Sum[DivisorSigma[1, k], {k, n}]; Table[a[n], {n, 51}] (* Jean-François Alcover, Dec 16 2011 *)
Accumulate[DivisorSigma[1,Range[60]]] (* Harvey P. Dale, Mar 13 2014 *)

A024916(n)=sum(k=1,n,n\k*k) \\ M. F. Hasler, Nov 22 2007

A024916(z) = { my(s,u,d,n,a,p); s = z*z; u = sqrtint(z); p = 2; for(d=1, u, n = z\d - z\(d+1); if(n<=1, p=d; break(), a = z%d; s -= (2*a+(n-1)*d)*n/2); ); u = z\p; for(d=2, u, s -= z%d); return(s); } \\ See the link for a nicely formatted version. - P. L. Patodia (pannalal(AT)usa.net), Jan 11 2008

A024916(n)={my(s=0,d=1,q=n);while(dPeter Polm, Aug 18 2014

A024916(n)={ my(s=n^2, r=sqrtint(n), nd=n, D); for(d=1, r, (1>=D=nd-nd=n\(d+1)) && (r=d-1) && break; s -= n%d*D+(D-1)*D\2*d); s - sum(d=2, n\(r+1), n%d)} \\ Slightly optimized version of Patodia's code. - M. F. Hasler, Apr 18 2015
(C#) See Polm link.

def A024916(n): return sum(k*(n//k) for k in range(1,n+1)) # Chai Wah Wu, Dec 17 2021

from math import isqrt
def A024916(n): return (-(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

[sum(sigma(k) for k in (1..n)) for n in (1..60)] # G. C. Greubel, Mar 15 2019

From Benoit Cloitre, Apr 28 2002: (Start)
a(n) = n^2 - A004125(n).
Asymptotically a(n) = n^2*Pi^2/12 + O(n*log(n)). (End)
G.f.: (1/(1-x))*Sum_{k>=1} x^k/(1-x^k)^2. - Benoit Cloitre, Apr 23 2003
a(n) = Sum_{m=1..n} (n - (n mod m)). - Roger L. Bagula and Gary W. Adamson, Oct 06 2006
a(n) = n^2*Pi^2/12 + O(n*log(n)^(2/3)) [Walfisz]. - Charles R Greathouse IV, Jun 19 2012
a(n) = A000217(n) + A153485(n). - Omar E. Pol, Jan 28 2014
a(n) = A000292(n) - A076664(n), n > 0. - Omar E. Pol, Feb 11 2014
a(n) = A078471(n) + A271342(n). - Omar E. Pol, Apr 08 2016
a(n) = (1/2)*(A222548(n) + A006218(n)). - Ridouane Oudra, Aug 03 2019
From Greg Dresden, Feb 23 2020: (Start)
a(n) = A092406(n) + 8, n>3.
a(n) = A160664(n) - 1, n>0. (End)
a(2*n) = A326123(n) + A326124(n). - Vaclav Kotesovec, Aug 18 2021
a(n) = Sum_{k=1..n} k * A010766(n,k). - Georg Fischer, Mar 04 2022

A072691 Decimal expansion of Pi^2/12.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 02 2002

			0.822467033424113218236207583323..

C. C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 98
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.11 p. 126 and section 8.5 p. 501.
Jolley, Summation of Series, Dover (1961) eq. (234) page 44.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Paul Bracken, Problem 4826, Crux Mathematicorum, Vol. 49, No. 3 (March, 2023), p. 157; Michel Bataille, Solution to Problem 4826, ibid., Vol. 49, No. 8 (Oct. 2023), p. 452.
Eugène-Charles Catalan, Mémoire sur la transformation des séries et sur quelques intégrales définies, Mémoires de l'Académie royale de Belgique, 1867, Vol. 33, pp. 1-50.
Brian Hawthorn, The Hardest Integral I've Ever Done, YouTube video, 2021.
Michael Penn, A viewer suggested integral, YouTube video, 2021.
Eric Weisstein's World of Mathematics, Dilogarithm
Index entries for transcendental numbers

Cf. A072692 (Pi^2/12 is in asymptotic formula related to sigma(n), A000203).
Cf. A113319 (sum_{i>=0} 1/(i^2+1)); A232883 (sum_{i>=0} 1/(2*i^2+1)).
Cf. A001008, A002805, A013661, A024916, A237593, A244583. Inverse of A377363.

RealDigits[Pi^2/12, 10, 105][[1]] (* Robert G. Wilson v *)

zeta(2)/2 \\ Michel Marcus, Sep 08 2014

-dilog(-1) \\ Charles R Greathouse IV, Apr 17 2015

Pi^2/12 \\ Charles R Greathouse IV, Apr 17 2015

sumnumrat(1/(2*x^2), 0) \\ Charles R Greathouse IV, Jan 20 2022

from mpmath import *
mp.dps=106
print([int(c) for c in list(str(zeta(2)/2))[2:-1]]) # Indranil Ghosh, Jul 08 2017

Equals 1/(1*2) + 1/(2*4) + 1/(3*6) + 1/(4*8) + ... [Jolley]
Equals -dilogarithm(-1). - Rick L. Shepherd, Jul 21 2004
Equals Sum_{n>=1} ((-1)^(n+1))/n^2 [Clawson]. - Alonso del Arte, Aug 15 2012
Equals Integral_{x=0..1} log((1+x^3)/(1-x^3))/x dx. - Bruno Berselli, May 13 2013
From Jean-François Alcover, May 17 2013: (Start)
Equals zeta(2)/2 = A013661/2.
Equals Integral_{x=1..2} log(x)/(x-1) dx. (End)
Equals lim_{n->infinity} A244583(n)/prime(n)^2. See A244583 for details. - Richard R. Forberg, Jan 04 2015
Equals Sum_{k>=1} H(k)/(k*2^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Aug 20 2020
Equals Integral_{0..infinity} x/(exp(x) + 1) dx. See Abramowitz-Stegun, 23.2.8, for s=2, p. 801. - Wolfdieter Lang, Sep 16 2020
Equals lim_{n->infinity} A024916(n)/(n^2). - Omar E. Pol, Dec 15 2021
Integral_{x=0..1} -log(x)/(x+1) dx. - Bernard Schott, Apr 25 2022
Equals 1/2 + Sum_{k>=1} H(k)/(k*(k+1)*(k+2)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Bracken, 2023). - Amiram Eldar, Oct 06 2023
Equals Integral_{x >= 0} x^2/cosh(x)^2 dx. - Peter Bala, Jun 20 2024
Equals 1 + (1/8)*Sum_{k >= 0} (-1)^(k-1) * (10*k + 13)/((k + 1)*(2*k + 1)^2*(2*k + 3)^2*binomial(2*k, k)). See Catalan, Section 35, equation 54. - Peter Bala, Aug 17 2024
Equals Integral_{x=0..oo} ((arctan(x) - Pi/4)*log(x^2 + 1))/(x^2) dx. - Kritsada Moomuang, Jun 04 2025

A046915 Sum of divisors of 10^n.

Original entry on oeis.org

1, 18, 217, 2340, 24211, 246078, 2480437, 24902280, 249511591, 2497558338, 24987792457, 249938963820, 2499694822171, 24998474116998, 249992370597277, 2499961853010960, 24999809265103951, 249999046325618058, 2499995231628286897

Offset: 0

Enoch Haga

A072692(n) = A049000(n) + a(n).

a(n) is the number of full-dimensional lattices in Z^(n+1) with volume 10. - Álvar Ibeas, Nov 29 2015

			At 10^1 the factors are 1, 2, 5, 10. The sum of these factors is 18: 1 + 2 + 5 + 10.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (18,-97,180,-100).

Cf. A000203 (sigma(n)), A049000, A072692. 10th row of A160870, shifted.

[1/4*(2^(n+1)-1)*(5^(n+1)-1): n in [0..20]]; // Vincenzo Librandi, Oct 03 2011

Table[DivisorSigma[1, 10^n], {n, 0, 18}] (* Jayanta Basu, Jun 30 2013 *)

Vec(-(10*x^2-1)/((x-1)*(2*x-1)*(5*x-1)*(10*x-1)) + O(x^100)) \\ Colin Barker, Jan 27 2015

a(n) = sigma(10^n); \\ Altug Alkan, Dec 04 2015

a(n) = 1/4*(2^(n+1)-1)*(5^(n+1)-1). E.g., a(1) = 1/4*(2^2-1)*(5^2-1) = 18. - Vladeta Jovovic, Dec 18 2001
a(n) = 18*a(n-1)-97*a(n-2)+180*a(n-3)-100*a(n-4). - Colin Barker, Jan 27 2015
G.f.: -(10*x^2-1) / ((x-1)*(2*x-1)*(5*x-1)*(10*x-1)). - Colin Barker, Jan 27 2015

A049000 Sum of sigma(j) for 1<=j<10^n, where sigma(j) is the sum of divisors of j.

Original entry on oeis.org

0, 69, 8082, 820741, 82231803, 8224494757, 822465638000, 82246686892516, 8224670172682646, 822467031614802290, 82246703327412473943, 8224670334073621455209, 822467033422857645316807, 82246703342395510922780776, 8224670334240978188556405240

Offset: 0

Enoch Haga and Jud McCranie

nonn

The ratio of successive terms approaches 100.

			For n=1, the sum of sigma(j) for j<10 is 1+3+4+7+6+12+8+15+13=69, so a(1)=69.

Amiram Eldar, Table of n, a(n) for n = 0..36 (calculated from the b-file at A072692)

Cf. A000203, A046915.
Cf. A072691 (Pi^2/12).
Cf. A072692 (sum for 1<=j<=10^n and other links).

a(n) = A072692(n) - A046915(n) ~ Pi^2/12 * 10^(2*n). - Amiram Eldar, Feb 16 2020

One more term from Rick L. Shepherd, Jul 03 2002

More terms from Amiram Eldar, Feb 16 2020

A136363 a(n) = Sum_{ composite k, 1 <= k <= 10^n} (Sum of divisors d of k with 1 < d < k).

Original entry on oeis.org

23, 3150, 321582, 32241015, 3224590836, 322466618438, 32246696794797, 3224670272194238, 322467032612360629, 32246703337400266401, 3224670334173560419030, 322467033423857340138979, 32246703342405509396897775, 3224670334241078180927002518, 322467033424112509326065894640

Offset: 1

Enoch Haga, Dec 26 2007, Dec 29 2007

nonn

			a(1) = 23 because the divisors through 10^1 are as follows: 4 (2); 6 (2,3); 8 (2,4); 9 (3); 10 (2,5). The sum is 2 + 2 + 3 + 2 + 4 + 3 + 2 + 5 = 23.

Amiram Eldar, Table of n, a(n) for n = 1..36 (calculated using the b-file at A072692)

Cf. A072692, A136364.

a(n) = A072692(n) - 10^n*(10^n+3)/2 + 1. - Max Alekseyev, May 10 2009

a(n) ~ (Pi^2/12 - 1/2) * 10^(2*n). - Amiram Eldar, Aug 05 2024

Edited by N. J. A. Sloane, Jan 12 2008
Extended by Max Alekseyev, May 10 2009
a(13)-a(15) from Amiram Eldar, Aug 05 2024

A188138 Sum of sigma_2(k) for 1 <= k <= 10^n, where sigma_2(k) is the sum of the divisors of k squared.

Original entry on oeis.org

1, 469, 407819, 401382971, 400757638164, 400692683389101, 400686363385965077, 400685705322499946270, 400685641565621401132515, 400685635084923815073475174, 400685634458741808360827818508, 400685634393583522561137962683069

Offset: 0

Robert G. Wilson v, Apr 25 2011

nonn

a(n) ~ 10^(3n)*zeta(3)/3.

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..18

Cf. A072692.

k = 1; lst = {}; s = 0; Do[ While[k < 10^n + 1, s = s + DivisorSigma[2, k]; k++]; AppendTo[lst, s], {n, 0, 9}]; lst
a[n_] := With[{nn=10^n}, Sum[Floor[nn/k]*k^2, {k, nn}]]; Array[a,9,0] (* T. D. Noe, Apr 25 2011 *)

a(10)-a(11) from Hiroaki Yamanouchi, Jul 06 2014

A189020 a(n) = Sum_{k=1..10^n} tau_4(k), where tau_4 is the number of ordered factorizations into 4 factors (A007426).

Original entry on oeis.org

1, 89, 3575, 93237, 1951526, 35270969, 578262093, 8840109380, 128217432396, 1784942188189, 24045237260214, 315312623543840, 4042957241191810, 50862246063060180, 629513636928477232, 7681900592647818929, 92587253467765253144, 1103781870246459696784, 13031388731053572679450, 152516435040764735691556, 1771079109308495896176156

Offset: 0

Andrew Lelechenko, Apr 15 2011

nonn

Using that tau_4 = tau_2 ** tau_2, where ** means Dirichlet convolution and tau_2 is (A000005), one can calculate a(n) faster than in O(10^n) operations - namely in O(10^(3n/4)) or even in O(10^(2n/3)). See links for details.

A. V. Lelechenko, The summation of the multiplicative functions (in Russian).

Cf. A057494 - partial sums up to 10^n of the divisors function tau_2 (A000005), A180361 - of the unitary divisors function tau_2* (A034444), A180365 - of the 3-divisors function tau_3 (A007425).

Also see A072692 for such sums of the sum of divisors function (A000203), A084237 for sums of Moebius function (A008683), A064018 for sums of Euler totient function (A000010).

a(n) = A061202(10^n) = Sum_{k=1..10^n} A007426(n).

a(16)-a(20) from Henri Lifchitz, Feb 05 2025

cp's OEIS Frontend

A024916 a(n) = Sum_{k=1..n} k*floor(n/k); also Sum_{k=1..n} sigma(k) where sigma(n) = sum of divisors of n (A000203).

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A072691 Decimal expansion of Pi^2/12.

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A046915 Sum of divisors of 10^n.

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A049000 Sum of sigma(j) for 1<=j<10^n, where sigma(j) is the sum of divisors of j.

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A136363 a(n) = Sum_{ composite k, 1 <= k <= 10^n} (Sum of divisors d of k with 1 < d < k).

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A188138 Sum of sigma_2(k) for 1 <= k <= 10^n, where sigma_2(k) is the sum of the divisors of k squared.

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