A160664 -id:A160664 - 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

A244051 Triangle read by rows in which row n lists the parts of the partitions of n into equal parts, in nonincreasing order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 08 2014

Row n has length sigma(n) = A000203(n).
Row sums give n*A000005(n) = A038040(n).
Column 1 is A000027.
Both columns 2 and 3 are A032742, n > 1.
For any k > 0 and t > 0, the sequence contains exactly one run of k consecutive t's. - Rémy Sigrist, Feb 11 2019
From Omar E. Pol, Dec 04 2019: (Start)
The number of parts congruent to 0 (mod m) in row m*n equals sigma(n) = A000203(n).
The number of parts greater than 1 in row n equals A001065(n), the sum of aliquot parts of n.
The number of parts greater than 1 and less than n in row n equals A048050(n), the sum of divisors of n except for 1 and n.
The number of partitions in row n equals A000005(n), the number of divisors of n.
The number of partitions in row n with an odd number of parts equals A001227(n).
The sum of odd parts in row n equals the sum of parts of the partitions in row n that have an odd number of parts, and equals the sum of all parts in the partitions of n into consecutive parts, and equals A245579(n) = n*A001227(n).
The decreasing records in row n give the n-th row of A056538.
Row n has n 1's which are all at the end of the row.
First n rows contain A000217(n) 1's.
The number of k's in row n is A126988(n,k).
The number of odd parts in row n is A002131(n).
The k-th block in row n has A027750(n,k) parts.
Right border gives A000012. (End)
The r-th row of the triangle begins at index k = A160664(r-1). - Samuel Harkness, Jun 21 2022

			Triangle begins:
   [1];
   [2], [1,1];
   [3], [1,1,1];
   [4], [2,2], [1,1,1,1];
   [5], [1,1,1,1,1];
   [6], [3,3], [2,2,2], [1,1,1,1,1,1];
   [7], [1,1,1,1,1,1,1];
   [8], [4,4], [2,2,2,2], [1,1,1,1,1,1,1,1];
   [9], [3,3,3], [1,1,1,1,1,1,1,1,1];
  [10], [5,5], [2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1];
  [11], [1,1,1,1,1,1,1,1,1,1,1];
  [12], [6,6], [4,4,4], [3,3,3,3], [2,2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1,1,1];
  [13], [1,1,1,1,1,1,1,1,1,1,1,1,1];
  [14], [7,7], [2,2,2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1,1,1,1,1];
  [15], [5,5,5], [3,3,3,3,3], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1];
  [16], [8,8], [4,4,4,4], [2,2,2,2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1];
  ...
For n = 6 the 11 partitions of 6 are [6], [3, 3], [4, 2], [2, 2, 2], [5, 1], [3, 2], [4, 1, 1], [2, 2, 1, 1], [3, 1, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1]. There are only four partitions of 6 that contain equal parts so the 6th row of triangle is [6], [3, 3], [2, 2, 2], [1, 1, 1, 1, 1, 1]. The number of parts equals sigma(6) = A000203(6) = 12. The row sum is A038040(6) = 6*A000005(6) = 6*4 = 24.
From _Omar E. Pol_, Dec 04 2019: (Start)
The structure of the above triangle is as follows:
   1;
   2 11;
   3    111;
   4 22     1111;
   5             11111;
   6 33 222            111111;
   7                          1111111;
   8 44     2222                      11111111;
   9    333                                    111111111;
  ... (End)

Samuel Harkness, Table of n, a(n) for n = 1..10000
Samuel Harkness, Log-log Scatterplot of the first 1000000 terms

Cf. A000005, A000012, A000041, A000203, A000217, A001065, A001227, A002131, A027750, A032742, A038040, A048050, A056538, A126988, A160664, A237593, A245579, A299765, A328365, A309400 (mirror).

A244051row[n_]:=Flatten[Map[ConstantArray[#,n/#]&,Reverse[Divisors[n]]]];
Array[A244051row,10] (* Paolo Xausa, Oct 16 2023 *)

tabf(nn) = {for (n=1, nn, d = Vecrev(divisors(n)); for (i=1, #d, for (j=1, n/d[i], print1(d[i], ", "));); print(););} \\ Michel Marcus, Nov 08 2014

A092406 a(1)=1, a(n) = sigma(n) if sigma(n) >= a(n-1), otherwise a(n) = a(n-1) + sigma(n).

Original entry on oeis.org

1, 3, 4, 7, 13, 25, 33, 48, 61, 79, 91, 119, 133, 157, 181, 212, 230, 269, 289, 331, 363, 399, 423, 483, 514, 556, 596, 652, 682, 754, 786, 849, 897, 951, 999, 1090, 1128, 1188, 1244, 1334, 1376, 1472, 1516, 1600, 1678, 1750, 1798, 1922, 1979, 2072, 2144, 2242

Offset: 1

Jon Perry, Mar 22 2004

nonn

Does a(n)=sigma(n) only for n=1,2,3,4?

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A024916, A160664.

nxt[{n_,a_}]:=Module[{s=DivisorSigma[1,n+1]},{n+1,If[s>=a,s,a+s]}]; NestList[ nxt,{1,1},60][[All,2]] (* Harvey P. Dale, Aug 04 2022 *)

{ v=vector(60); v[1]=sigma(1); for (i=2,60, if (sigma(i)

from math import isqrt
def A092406(n): return (-(s:=isqrt(n))**2*(s+1) + sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1)-8 if n>3 else (1 if n<2 else n+1) # Chai Wah Wu, Oct 22 2023

a(n) = A024916(n) - 8 and A160664(n) - 9, for n > 3. - Greg Dresden, Feb 23 2020

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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A244051 Triangle read by rows in which row n lists the parts of the partitions of n into equal parts, in nonincreasing order.

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A092406 a(1)=1, a(n) = sigma(n) if sigma(n) >= a(n-1), otherwise a(n) = a(n-1) + sigma(n).

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