A123229 -id:A123229 - 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

A127093 Triangle read by rows: T(n,k)=k if k is a divisor of n; otherwise, T(n,k)=0 (1 <= k <= n).

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 5, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 0, 0, 7, 1, 2, 0, 4, 0, 0, 0, 8, 1, 0, 3, 0, 0, 0, 0, 0, 9, 1, 2, 0, 0, 5, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 12, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1, 2, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 14

Offset: 1

Gary W. Adamson, Jan 05 2007, Apr 04 2007

Sum of terms in row n = sigma(n) (sum of divisors of n).
Euler's derivation of A127093 in polynomial form is in his proof of the formula for Sigma(n): (let S=Sigma, then Euler proved that S(n) = S(n-1) + S(n-2) - S(n-5) - S(n-7) + S(n-12) + S(n-15) - S(n-22) - S(n-26), ...).
[Young, pp. 365-366], Euler begins, s = (1-x)*(1-x^2)*(1-x^3)*... = 1 - x - x^2 + x^5 + x^7 - x^12 ...; log s = log(1-x) + log(1-x^2) + log(1-x^3) ...; differentiating and then changing signs, Euler has t = x/(1-x) + 2x^2/(1-x^2) + 3x^3/(1-x^3) + 4x^4/(1-x^4) + 5x^5/(1-x^5) + ...
Finally, Euler expands each term of t into a geometric series, getting A127093 in polynomial form: t =
  x +  x^2 +  x^3 +  x^4 +  x^5 +  x^6 +  x^7 +  x^8 + ...
    + 2x^2        + 2x^4        + 2x^6        + 2x^8 + ...
           + 3x^3               + 3x^6               + ...
                  + 4x^4                      + 4x^8 + ...
                         + 5x^5                      + ...
                                + 6x^6               + ...
                                       + 7x^7        + ...
                                              + 8x^8 + ...
T(n,k) is the sum of all the k-th roots of unity each raised to the n-th power. - Geoffrey Critzer, Jan 02 2016
From Davis Smith, Mar 11 2019: (Start)
For n > 1, A020639(n) is the leftmost term, other than 0 or 1, in the n-th row of this array. As mentioned in the Formula section, the k-th column is period k: repeat [k, 0, 0, ..., 0], but this also means that it's the characteristic function of the multiples of k multiplied by k. T(n,1) = A000012(n), T(n,2) = 2*A059841(n), T(n,3) = 3*A079978(n), T(n,4) = 4*A121262(n), T(n,5) = 5*A079998(n), and so on.
The terms in the n-th row, other than 0, are the factors of n. If n > 1 and for every k, 1 <= k < n, T(n,k) = 0 or 1, then n is prime. (End)
From Gary W. Adamson, Aug 07 2019: (Start)
Row terms of the triangle can be used to calculate E(n) in A002654): (1, 1, 0, 1, 2, 0, 0, 1, 1, 2, ...), and A004018, the number of points in a square lattice on the circle of radius sqrt(n), A004018: (1, 4, 4, 0, 4, 8, 0, 0, 4, ...).
As to row terms in the triangle, let E(n) of even terms = 0,
  E(integers of the form 4*k - 1 = (-1), and E(integers of the form 4*k + 1 = 1.
Then E(n) is the sum of the E(n)'s of the factors of n in the triangle rows.  Example: E(10) = Sum: ((E(1) + E(2) + E(5) + E(10)) = ((1 + 0 + 1 + 0) = 2, matching A002654(10).
To get A004018, multiply the result by 4, getting A004018(10) = 8.
The total numbers of lattice points = 4r^2 = E(1) + ((E(2))/2 + ((E(3))/3 + ((E(4))/4 + ((E(5))/5 + .... Since E(even integers) are zero, E(integers of the form (4*k - 1)) = (-1), and E(integers of the form (4*k + 1)) = (+1); we are left with 4r^2 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ..., which is approximately equal to Pi(r^2). (End)
T(n,k) is also the number of parts in the partition of n into k equal parts. - Omar E. Pol, May 05 2020

			T(8,4) = 4 since 4 divides 8.
T(9,3) = 3 since 3 divides 9.
First few rows of the triangle:
  1;
  1, 2;
  1, 0, 3;
  1, 2, 0, 4;
  1, 0, 0, 0, 5;
  1, 2, 3, 0, 0, 6;
  1, 0, 0, 0, 0, 0, 7;
  1, 2, 0, 4, 0, 0, 0, 8;
  1, 0, 3, 0, 0, 0, 0, 0, 9;
  ...

David Wells, "Prime Numbers, the Most Mysterious Figures in Math", John Wiley & Sons, 2005, appendix.
L. Euler, "Discovery of a Most Extraordinary Law of the Numbers Concerning the Sum of Their Divisors"; pp. 358-367 of Robert M. Young, "Excursions in Calculus, An Interplay of the Continuous and the Discrete", MAA, 1992. See p. 366.

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Grant Sanderson, Pi hiding in prime regularities
Leonhard Euler, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs, 1747, The Euler Archive, (Eneström Index) E175.
Leonhard Euler, Observatio de summis divisorum
Eric Weisstein's World of Mathematics, Divisor

Reversal = A127094
Cf. A127094, A123229, A127096, A127097, A127098, A127099, A000203, A126988, A127013, A127057, A038040, A024916, A060640, A001001.
Cf. A000005, A060640.
Cf. A027750.
Cf. A000012 (the first column), A020639, A059841 (the second column when multiplied by 2), A079978 (the third column when multiplied by 2), A079998 (the fifth column when multiplied by 5), A121262 (the fourth column when multiplied by 4).
Cf. A002654, A004018, A008683, A050873.

mod(row()-1;column()) - mod(row();column()) + 1 - Mats Granvik, Aug 31 2007

a127093 n k = a127093_row n !! (k-1)
a127093_row n = zipWith (*) [1..n] $ map ((0 ^) . (mod n)) [1..n]
a127093_tabl = map a127093_row [1..]
-- Reinhard Zumkeller, Jan 15 2011

A127093:=proc(n,k) if type(n/k, integer)=true then k else 0 fi end:
for n from 1 to 16 do seq(A127093(n,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Jan 20 2007

t[n_, k_] := k*Boole[Divisible[n, k]]; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 17 2014 *)
Table[ SeriesCoefficient[k*x^k/(1 - x^k), {x, 0, n}], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 14 2015 *)

trianglerows(n) = for(x=1, n, for(k=1, x, if(x%k==0, print1(k, ", "), print1("0, "))); print(""))
/* Print initial 9 rows of triangle as follows: */
trianglerows(9) \\ Felix Fröhlich, Mar 26 2019

k-th column is composed of "k" interspersed with (k-1) zeros.
Let M = A127093 as an infinite lower triangular matrix and V = the harmonic series as a vector: [1/1, 1/2, 1/3, ...]. then M*V = d(n), A000005: [1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ...]. M^2 * V = A060640: [1, 5, 7, 17, 11, 35, 15, 49, 34, 55, ...]. - Gary W. Adamson, May 10 2007
T(n,k) = ((n-1) mod k) - (n mod k) + 1 (1 <= k <= n). - Mats Granvik, Aug 31 2007
T(n,k) = k * 0^(n mod k). - Reinhard Zumkeller, Jan 15 2011
G.f.: Sum_{k>=1} k * x^k * y^k/(1-x^k) = Sum_{m>=1} x^m * y/(1 - x^m*y)^2. - Robert Israel, Aug 08 2016
T(n,k) = Sum_{d|k} mu(k/d)*sigma(gcd(n,d)). - Ridouane Oudra, Apr 05 2025

A127096 Triangle T(n,m) = A000012*A127094 read by rows.

Original entry on oeis.org

1, 3, 1, 6, 1, 1, 10, 1, 3, 1, 15, 1, 3, 1, 1, 21, 1, 3, 4, 3, 1, 28, 1, 3, 4, 3, 1, 1, 36, 1, 3, 4, 7, 1, 3, 1, 45, 1, 3, 4, 7, 1, 6, 1, 1, 55, 1, 3, 4, 7, 6, 6, 1, 3, 1, 66, 1, 3, 4, 7, 6, 6, 1, 3, 1, 1, 78, 1, 3, 4, 7, 6, 12, 1, 7, 4, 3, 1, 91, 1, 3, 4, 7, 6, 12, 1, 7, 4, 3, 1, 1, 105, 1, 3, 4, 7, 6, 12, 8, 7, 4, 3, 1, 3, 1

Offset: 1

Gary W. Adamson, Jan 05 2007

Consider A000012 as a lower-left all-1's triangle, and build the matrix product by multiplication with A127094 from the right.

			First few rows of the triangle are:
   1;
   3, 1,
   6, 1, 1;
  10, 1, 3, 1;
  15, 1, 3, 1, 1;
  21, 1, 3, 4, 3, 1;
  28, 1, 3, 4, 3, 1, 1;
  ...

Cf. A127093, A127094, A123229, A024916 (row sums), A000203, A126988.

A127093 := proc(n,m) if n mod m = 0 then m; else 0 ; fi; end:
A127094 := proc(n,m) A127093(n, n-m+1) ; end:
A127096 := proc(n,m) add( A127094(j,m),j=m..n) ; end:
for n from 1 to 15 do for m from 1 to n do printf("%d,",A127096(n,m)) ; od: od: # R. J. Mathar, Aug 18 2009

T[n_, m_] := Sum[1 + Mod[j, m - j - 1] - Mod[1 + j, m - j - 1], {j, m, n}];
Table[T[n, m], {n, 1, 14}, {m, 1, n}] // Flatten (* Jean-François Alcover, Sep 15 2023 *)

T(n,m) = Sum_{j=m..n} A000012(n,j)*A127094(j,m) = Sum_{j=m..n} A127094(j,m).

Edited and extended by R. J. Mathar, Aug 18 2009

A127099 Triangle T(n,m) = A126988 *A127093 read by rows.

Original entry on oeis.org

1, 3, 2, 4, 0, 3, 7, 6, 0, 4, 6, 0, 0, 0, 5, 12, 8, 9, 0, 0, 6, 8, 0, 0, 0, 0, 0, 7, 15, 14, 0, 12, 0, 0, 0, 8, 13, 0, 12, 0, 0, 0, 0, 0, 9, 18, 12, 0, 0, 15, 0, 0, 0, 0, 10, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 28, 24, 21, 16, 0, 18, 0, 0, 0, 0, 0, 12, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 24, 16, 0, 0

Offset: 1

Gary W. Adamson, Jan 05 2007

Multiply the infinite lower triangular matrices A126988 and A127093.

			First few rows of the triangle are:
1;
3, 2;
4, 0, 3;
7, 6, 0, 4;
6, 0, 0, 0, 5;
12, 8, 9, 0, 0, 6;
8, 0, 0, 0, 0, 0, 7;
15, 14, 0, 12, 0, 0, 0, 8;
13, 0, 12, 0, 0, 0, 0, 0, 9;
18, 12, 0, 0, 15, 0, 0, 0, 0, 10;
...

Cf. A060640 (row sums), A000203, A127093, A127094, A123229, A127096, A127097, A127098, A127013, A127057.

T(n,m) = sum_{j=m..n} A126988(n,j)*A127093(j,m).

T(n,1) = A000203(n).

Extended by R. J. Mathar, Aug 18 2009

A127108 Triangle read by rows, A127099 * A000012.

Original entry on oeis.org

1, 5, 2, 7, 3, 3, 17, 10, 4, 4, 11, 5, 5, 5, 5, 35, 23, 15, 6, 6, 6, 15, 7, 7, 7, 7, 7, 7, 49, 34, 20, 20, 8, 8, 8, 8, 34, 21, 21, 9, 9, 9, 9, 9, 9, 55, 37, 25, 25, 25, 10, 10, 10, 10, 10, 23, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 119, 91, 67, 46, 30, 30, 12, 12, 12, 12, 12, 12

Offset: 0

Gary W. Adamson, Jan 05 2007, Jul 27 2008

The operation A000012 * A127099 generates n-th row of the triangle by taking partial sums of n-th row of triangle A127099. Row 4 of A127099 (7, 6, 0, 4) becomes row 4 of A127108: (17, 10, 4, 4).
Row sums = A001001: (1, 7, 13, 35, 31, 91, ...).
Left column of the triangle = A060640: (1, 5, 7, 17, 11, 35, ...).

			First few rows of the triangle:
   1;
   5,  2;
   7,  3,  3;
  17, 10,  4,  4;
  11,  5,  5,  5,  5;
  35, 23, 15,  6,  6,  6;
  15,  7,  7,  7,  7,  7,  7;
  49, 34, 20, 20,  8,  8,  8,  8;
  34, 21, 21,  9,  9,  9,  9,  9,  9;
  55, 37, 25, 25, 25, 10, 10, 10, 10, 10;
  ...

Cf. A060640, A001001, A126988, A127093, A000203, A127094, A123229, A127096, A127098, A127099.

Triangle read by rows, A127099 * A000012.

Edited by N. J. A. Sloane, Aug 13 2008 at the suggestion of R. J. Mathar

cp's OEIS Frontend

A127095 Duplicate of A123229.

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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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A127093 Triangle read by rows: T(n,k)=k if k is a divisor of n; otherwise, T(n,k)=0 (1 <= k <= n).

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A127096 Triangle T(n,m) = A000012*A127094 read by rows.

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A127099 Triangle T(n,m) = A126988 *A127093 read by rows.

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A127108 Triangle read by rows, A127099 * A000012.

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