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

1,2

Comments

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

References

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

Links

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.

Formula

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

Example

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)

Maple

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

Mathematica

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 *)

Prog

(PARI) A024916(n)=sum(k=1, n, n\k*k) \\ M. F. Hasler, Nov 22 2007
(PARI) 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
(PARI) A024916(n)={my(s=0, d=1, q=n); while(d<q, s+=q*(q+1+2*d)\2; d++; q=n\d; ); return(s-d*(d-1)\2*d+q*(q+1)\2); } \\ Peter Polm, Aug 18 2014
(PARI) 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.
(Haskell)
a024916 n = sum $ map (\k -> k * div n k) [1..n]
-- Reinhard Zumkeller, Apr 20 2015
(Magma) [(&+[DivisorSigma(1, k): k in [1..n]]): n in [1..60]]; // G. C. Greubel, Mar 15 2019
(Sage) [sum(sigma(k) for k in (1..n)) for n in (1..60)] # G. C. Greubel, Mar 15 2019
(Python)
def A024916(n): return sum(k*(n//k) for k in range(1, n+1)) # Chai Wah Wu, Dec 17 2021

Crossrefs

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.

Sequence in context: A071422 A212538 A113902 * A212539 A102216 A001182

Adjacent sequences: A024913 A024914 A024915 * A024917 A024918 A024919

Keyword

nonn,nice

Author

Clark Kimberling

Status

approved