A024916 -id:A024916 - cp's OEIS frontend

A307042 Partial sums of the exponential divisors sum function: Sum_{k=1..n} esigma(k), where esigma is A051377.

1, 3, 6, 12, 17, 23, 30, 40, 52, 62, 73, 91, 104, 118, 133, 155, 172, 196, 215, 245, 266, 288, 311, 341, 371, 397, 427, 469, 498, 528, 559, 593, 626, 660, 695, 767, 804, 842, 881, 931, 972, 1014, 1057, 1123, 1183, 1229, 1276, 1342, 1398, 1458, 1509, 1587, 1640

Offset: 1

Amiram Eldar, Mar 21 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. Fabrykowski and M. V. Subbarao, The maximal order and the average order of multiplicative function sigma^(e)(n), in Théorie des nombres/Number theory (Quebec, PQ, 1987), 201-206, de Gruyter, Berlin, 1989.

Cf. A024916, A051377, A064609, A275480.

esigma[n_] := Times @@ (Sum[ First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; Accumulate[Array[esigma, 60]] (* after Jean-François Alcover at A051377 *)

a(n) ~ B * n^2, where B = 0.5682854937... (A275480).

A346533 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000203 together with the sum of A000203(n-1) and A000203(n), with a(1) = 1.

Original entry on oeis.org

1, 4, 1, 7, 1, 3, 11, 1, 3, 4, 13, 1, 3, 4, 7, 18, 1, 3, 4, 7, 6, 20, 1, 3, 4, 7, 6, 12, 23, 1, 3, 4, 7, 6, 12, 8, 28, 1, 3, 4, 7, 6, 12, 8, 15, 31, 1, 3, 4, 7, 6, 12, 8, 15, 13, 30, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 40, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 42, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 38

Offset: 1

Omar E. Pol, Jul 22 2021

T(n,k) is the total area (or number of cells) of the terraces that are in the k-th level that contains terraces starting from the top of the symmetric tower (a polycube) described in A221529.
The height of the tower equals A000041(n-1).
The terraces of the tower are the symmetric representation of sigma.
The terraces are in the levels that are the partition numbers A000041 starting from the base.
Note that for n >= 2 there are n - 1 terraces because the lower terrace of the tower is formed by two symmetric representations of sigma in the same level.

			Triangle begins:
  1;
  4;
  1, 7;
  1, 3, 11;
  1, 3,  4, 13;
  1, 3,  4,  7, 18;
  1, 3,  4,  7,  6, 20;
  1, 3,  4,  7,  6, 12, 23;
  1, 3,  4,  7,  6, 12,  8, 28;
  1, 3,  4,  7,  6, 12,  8, 15, 31;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 30;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 40;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 12, 42;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 12, 28, 38;
  ...
For n = 7, sigma(7) = 1 + 7 = 8 and sigma(6) = 1 + 2 + 3 + 6 = 12, and 8 + 12 = 20, so the last term of row 7 is T(7,6) = 20. The other terms in row 7 are the first five terms of A000203, so the 7th row of the triangle is [1, 3, 4, 7, 6, 20].
For n = 7 we can see below the top view and the lateral view of the pyramid described in A245092 (with seven levels) and the top view and the lateral view of the tower described in A221529 (with 11 levels).
                                           _
                                          | |
                                          | |
                                          | |
        _                                 |_|_
       |_|_                               |   |
       |_ _|_                             |_ _|_
       |_ _|_|_                           |   | |
       |_ _ _| |_                         |_ _|_|_
       |_ _ _|_ _|_                       |_ _ _| |_
       |_ _ _ _| | |_                     |_ _ _|_ _|_ _
       |_ _ _ _|_|_ _|                    |_ _ _ _|_|_ _|
.
         Figure 1.                           Figure 2.
        Lateral view                       Lateral view
       of the pyramid.                     of the tower.
.
.       _ _ _ _ _ _ _                      _ _ _ _ _ _ _
       |_| | | | | | |                    |_| | | | |   |
       |_ _|_| | | | |                    |_ _|_| | |   |
       |_ _|  _|_| | |                    |_ _|  _|_|   |
       |_ _ _|    _|_|                    |_ _ _|    _ _|
       |_ _ _|  _|                        |_ _ _|  _|
       |_ _ _ _|                          |       |
       |_ _ _ _|                          |_ _ _ _|
.
          Figure 3.                          Figure 4.
          Top view                           Top view
       of the pyramid.                     of the tower.
.
Both polycubes have the same base which has an area equal to A024916(7) = 41 equaling the sum of the 7th row of triangle.
Note that in the top view of the tower the symmetric representation of sigma(6) and the symmetric representation of sigma(7) appear unified in the level 1 of the structure as shown above in the figure 4 (that is due the first two partition numbers A000041 are [1, 1]), so T(7,6) = sigma(7) + sigma(6) = 8 + 12 = 20.
.
Illustration of initial terms:
   Row 1    Row 2      Row 3      Row 4        Row 5          Row 6
.
    1        4         1 7        1 3 11       1 3 4 13       1 3 4 7 18
.   _        _ _       _ _ _      _ _ _ _      _ _ _ _ _      _ _ _ _ _ _
   |_|      |   |     |_|   |    |_| |   |    |_| | |   |    |_| | | |   |
            |_ _|     |    _|    |_ _|   |    |_ _|_|   |    |_ _|_| |   |
                      |_ _|      |      _|    |_ _|  _ _|    |_ _|  _|   |
                                 |_ _ _|      |     |        |_ _ _|    _|
                                              |_ _ _|        |        _|
                                                             |_ _ _ _|
.

Paolo Xausa, Table of n, a(n) for n = 1..11176 (rows 1..150 of the triangle, flattened)
Omar E. Pol, Illustration of the partitions of 10, prism and tower

Mirror of A340584.
The length of row n is A028310(n-1).
Row sums give A024916.
Leading diagonal gives A092403.
Other diagonals give A000203.
Companion of A346562.
Cf. A175254 (volume of the pyramid).
Cf. A066186 (volume of the tower).
Cf. A000041, A221529, A237270, A237593, A245092, A245093 (similar), A336811, A336812, A338156, A339106, A340035.

A346533row[n_]:=If[n==1,{1},Join[DivisorSigma[1,Range[n-2]],{Total[DivisorSigma[1,{n-1,n}]]}]];Array[A346533row,15] (* Paolo Xausa, Oct 23 2023 *)

A350106 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j * floor(n/j)^k.

Original entry on oeis.org

1, 1, 4, 1, 6, 8, 1, 10, 14, 15, 1, 18, 32, 31, 21, 1, 34, 86, 87, 45, 33, 1, 66, 248, 295, 153, 81, 41, 1, 130, 734, 1095, 669, 309, 101, 56, 1, 258, 2192, 4231, 3201, 1521, 443, 150, 69, 1, 514, 6566, 16647, 15765, 8373, 2633, 722, 191, 87, 1, 1026, 19688, 66055, 78393, 48321, 17411, 4746, 1005, 253, 99

Offset: 1

Seiichi Manyama, Dec 14 2021

			Square array begins:
   1,   1,   1,    1,     1,      1,      1, ...
   4,   6,  10,   18,    34,     66,    130, ...
   8,  14,  32,   86,   248,    734,   2192, ...
  15,  31,  87,  295,  1095,   4231,  16647, ...
  21,  45, 153,  669,  3201,  15765,  78393, ...
  33,  81, 309, 1521,  8373,  48321, 284709, ...
  41, 101, 443, 2633, 17411, 119321, 828323, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=1..3 give A024916, A350107, A350108.
T(n,n) gives A350109.
Cf. A319649, A344725.

T[n_, k_] := Sum[j * Floor[n/j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Dec 14 2021 *)

PARI
```
T(n, k) = sum(j=1, n, j*(n\j)^k);
```

T(n, k) = sum(j=1, n, j*sumdiv(j, d, (d^k-(d-1)^k)/d));

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} (j^k - (j - 1)^k) * x^j/(1 - x^j)^2.

T(n,k) = Sum_{j=1..n} j * Sum_{d|j} (d^k - (d - 1)^k)/d.

A365409 a(n) = Sum_{k=1..n} binomial(floor(n/k)+3,4).

Original entry on oeis.org

1, 6, 17, 42, 78, 149, 234, 379, 555, 815, 1102, 1557, 2013, 2662, 3388, 4349, 5319, 6695, 8026, 9846, 11712, 14027, 16328, 19503, 22464, 26200, 30030, 34759, 39255, 45221, 50678, 57623, 64465, 72579, 80469, 90665, 99805, 111020, 122146, 135566, 147908, 163638

Offset: 1

Seiichi Manyama, Oct 23 2023

Partial sums of A059358.

Cf. A006218, A024916, A064602, A064603, A364970, A365439.

a(n) = sum(k=1, n, binomial(n\k+3, 4));

from math import isqrt, comb
def A365409(n): return -(s:=isqrt(n))**2*comb(s+3,3)+sum((q:=n//k)*((comb(k+2,3)<<2)+comb(q+3,3)) for k in range(1,s+1))>>2 # Chai Wah Wu, Oct 26 2023

a(n) = Sum_{k=1..n} binomial(k+2,3) * floor(n/k).
G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^4 = 1/(1-x) * Sum_{k>=1} binomial(k+2,3) * x^k/(1-x^k).
a(n) = (A064603(n)+3*A064602(n)+2*A024916(n))/6. - Chai Wah Wu, Oct 26 2023

A024917 a(n) = Sum_{k=2..n} k*floor(n/k).

Original entry on oeis.org

2, 5, 11, 16, 27, 34, 48, 60, 77, 88, 115, 128, 151, 174, 204, 221, 259, 278, 319, 350, 385, 408, 467, 497, 538, 577, 632, 661, 732, 763, 825, 872, 925, 972, 1062, 1099, 1158, 1213, 1302, 1343, 1438, 1481, 1564, 1641, 1712, 1759, 1882, 1938, 2030, 2101, 2198, 2251

Offset: 2

Clark Kimberling

nonn

Cf. A002541, A024916, A345682.

[&+[k*Floor(n/k): k in [2..n]]: n in [2..55]]; // Bruno Berselli, Jan 08 2012

Table[Sum[k*Floor[n/k],{k,2,n}],{n,2,60}] (* Harvey P. Dale, Mar 13 2015 *)

a(n) = sum(k=2,n, k*floor(n/k)); \\ Michel Marcus, Sep 02 2019

from math import isqrt
def A024917(n): return (-(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1)-n # Chai Wah Wu, Oct 23 2023

G.f.: (1/(1 - x)) * Sum_{k>=2} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 02 2019

A086718 Convolution of sequence of primes with sequence sigma(n).

Original entry on oeis.org

2, 9, 22, 48, 85, 151, 231, 355, 500, 709, 937, 1267, 1617, 2069, 2575, 3193, 3860, 4686, 5549, 6593, 7725, 8985, 10337, 11961, 13591, 15464, 17498, 19714, 22036, 24690, 27378, 30382, 33603, 37023, 40597, 44733, 48720, 53152, 57950, 62978, 68074, 73898, 79558

Offset: 1

Jon Perry, Jul 29 2003

nonn

From Omar E. Pol, Dec 06 2021: (Start)
Antidiagonal sums of A272214.
Convolution of A000040 and A000203.
Convolution of A054541 and A024916.
Convolution of the nonzero terms of A007504 and A340793.
a(n) is also the volume of a tower or polycube in which the successive terraces are the symmetric representation of sigma(k), k = 1..n starting from the top, and the successive heights of the terraces are the prime numbers starting from the base. (End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A000203.

Cf. A007504, A024916, A054541, A066186, A175254, A191831, A237593, A272214, A340793.

N:= 100: # to get a(1)..a(N)
P:= [seq(ithprime(i),i=1..N+1)]:
S:= [seq(numtheory:-sigma(i),i=1..N+1)]:
seq(add(P[i]*S[n-i],i=1..n-1),n=2..N+1); # Robert Israel, Sep 09 2020

p=primes(30); s=vector(30,i, sigma(i)); conv(u,v)=local(w); w=vector(length(u),i,sum(j=1,i,u[j]*v[i+1-j])); w;
conv(p,s)

More terms from Robert Israel, Sep 09 2020

A123327 a(n) = A000203(n) + A004125(n).

Original entry on oeis.org

1, 3, 5, 8, 10, 15, 16, 23, 25, 31, 34, 45, 42, 55, 60, 67, 69, 86, 84, 103, 102, 113, 122, 145, 134, 154, 165, 180, 181, 210, 199, 230, 232, 251, 266, 289, 271, 308, 325, 348, 339, 380, 369, 412, 417, 430, 451, 498, 471, 513, 521, 552, 559, 612, 601, 640, 633

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Sep 26 2006; Juri-Stepan Gerasimov, Jul 02 2009

Another definition for this sequence: Let M be the matrix defined in A111490. Sequence gives M(1,1), M(1,2) + M(2,2), M(1,3) + M(2,3) + M(3,3), etc., i.e. a(n)= Sum_{i=1..n} M(i,n).
Proof from Hartmut F. W. Hoft, Feb 02 2014 that the two definitions agree: (Start)
For all n>=1 the following simplifications hold for the partial sums of the two sequences:
sum[1..n] a(k) = sum[1..n] A000203(k) + sum[1..n] A004125(k)
               = A024916(n) + sum[1..n] A004125(k)
               = n^2 + sum[1..n-1] A004125(k)
               = sum[1..n] A123327(k).
An inductive argument then shows that the two definitions agree.
(End)

			1(=1+0), 3(=3+0), 5(=4+1), 8(=7+1), 10(=6+4), 15(=12+3), 16(=8+8), etc.

Cf. A000203, A004125, A024916, A111490.

Lim=57;s2=Table[Sum[Mod[n, k], {k, 2, n-1}], {n, Lim}];Table[DivisorSigma[1, n]+s2[[n]],{n,Lim}] (* James C. McMahon, Nov 20 2024 *)

from math import isqrt
def A123327(n): return n**2+((s:=isqrt(n-1))**2*(s+1)-sum((q:=(n-1)//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) # Chai Wah Wu, Oct 22 2023

a(n) = A000290(n) - A024916(n-1), n > 1. - Omar E. Pol, Jan 29 2014

Corrected (83 replaced by 103) by R. J. Mathar, May 21 2010

Edited by N. J. A. Sloane, Feb 02 2014, merging A162383 from Juri-Stepan Gerasimov with the present sequence. Thanks to Omar E. Pol for noticing the duplication.

A134867 A010766 * A000012.

Original entry on oeis.org

1, 3, 1, 5, 2, 1, 8, 4, 2, 1, 10, 5, 3, 2, 1, 14, 8, 5, 3, 2, 1, 16, 9, 6, 4, 3, 2, 1, 20, 12, 8, 6, 4, 3, 2, 1, 23, 14, 10, 7, 5, 4, 3, 2, 1, 27, 17, 12, 9, 7, 5, 4, 3, 2, 1, 29, 18, 13, 10, 8, 6, 5, 4, 3, 2, 1, 35, 23, 17, 13, 10, 8, 6, 5, 4, 3, 2, 1, 37, 24, 18, 14, 11, 9, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, Nov 14 2007

			First few rows of the triangle:
   1;
   3,  1;
   5,  2,  1;
   8,  4,  2, 1;
  10,  5,  3, 2, 1;
  14,  8,  5, 3, 2, 1;
  16,  9,  6, 4, 3, 2, 1;
  20, 12,  8, 6, 4, 3, 2, 1;
  23, 14, 10, 7, 5, 4, 3, 2, 1;
  27, 17, 12, 9, 7, 5, 4, 3, 2, 1;
  ...

Column k=1..4 give: A006218, A002541, A366968, A366972.
Row sums give A024916.
Cf. A010766, A135539.

t = Table[Sum[Floor[n/h], {h, k, n}], {n, 0, 10}, {k, 1, n}];
u = Flatten[t]  (* A134867 array *)
TableForm[t]    (* A134867 sequence *)
(* Clark Kimberling, Oct 11 2014 *)

T(n, k) = sum(j=k, n, n\j); \\ Seiichi Manyama, Oct 30 2023

A010766 * A000012 as infinite lower triangular matrices.
Triangle read by rows, partial row sums of A010766 starting fromt the right.
G.f. of column k: 1/(1-x) * Sum_{j>=1} x^(k*j)/(1-x^j) = 1/(1-x) * Sum_{j>=k} x^j/(1-x^j). - Seiichi Manyama, Oct 30 2023

More terms from Seiichi Manyama, Oct 30 2023

A160664 a(n) = a(n-1) + A000203(n), a(0)=1.

Original entry on oeis.org

1, 2, 5, 9, 16, 22, 34, 42, 57, 70, 88, 100, 128, 142, 166, 190, 221, 239, 278, 298, 340, 372, 408, 432, 492, 523, 565, 605, 661, 691, 763, 795, 858, 906, 960, 1008, 1099, 1137, 1197, 1253, 1343, 1385, 1481, 1525, 1609, 1687, 1759, 1807, 1931, 1988, 2081, 2153

Offset: 0

Ctibor O. Zizka, May 22 2009

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A054519, A092406, A024916. - Greg Dresden, Feb 23 2020

ListTools:-PartialSums(map(numtheory:-sigma,[1,$1..100])); # Robert Israel, Dec 19 2016

lst = {1}; a = 1; Do[a = a + DivisorSigma[1, n]; AppendTo[lst, a], {n, 80}]; lst (* Carl Najafi, Aug 21 2011 *)
Transpose[NestList[{First[#]+1,Last[#]+DivisorSigma[1,First[#]+1]}&,{0,1},50]][[2]] (* Harvey P. Dale, May 05 2012 *)

a(n)=1+sum(k=1,n,sigma(k)) \\ Charles R Greathouse IV, Aug 22 2011

from math import isqrt
def A160664(n): return (-(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 22 2023

a(n) = 1 + A024916(n). - R. J. Mathar, May 25 2009

a(n) = 9 + A092406(n) for n>3. - Greg Dresden, Feb 23 2020

More terms from Carl Najafi, Aug 21 2011

A168012 a(n) = sum of all divisors of all numbers k such that n^2 <= k < (n+1)^2.

Original entry on oeis.org

8, 48, 133, 302, 516, 923, 1346, 2038, 2768, 3891, 4810, 6572, 7959, 10066, 12186, 14944, 17261, 21210, 23992, 28497, 32550, 37742, 42111, 48906, 54252, 61280, 68153, 76958, 82942, 94661, 101882, 113082, 123794, 135583, 145630, 161526

Offset: 1

Omar E. Pol, Nov 16 2009

nonn

			a(2) = 48 because the numbers k are 4,5,6,7 and 8 (since 2^2 <= k < 3^2) and sigma(4) + sigma(5) + sigma(6) + sigma(7) + sigma(8) = 7 + 6 + 12 + 8 + 15 = 48, where sigma(n) is the sum of divisors of n (see A000203).

Paolo Xausa, Table of n, a(n) for n = 1..1000

Cf. A000203, A024916, A168010, A168011, A168013.

A168012[n_]:=Sum[DivisorSigma[1,k],{k,n^2,(n+1)^2-1}];
Array[A168012,50] (* Paolo Xausa, Oct 23 2023 *)

a(n)=sum(k=n^2,(n+1)^2-1,sigma(k)) \\ Franklin T. Adams-Watters, May 14 2010

def A168012(n):
    a, b = n*(n+2),(n-1)*(n+1)
    return (sum((q:=a//k)*((s:=k<<1)+q+1)-(r:=b//k)*(s+r+1) for k in range(1,n))>>1)+5*n+3 # Chai Wah Wu, Oct 23 2023

More terms from Franklin T. Adams-Watters, May 14 2010

cp's OEIS Frontend

A307042 Partial sums of the exponential divisors sum function: Sum_{k=1..n} esigma(k), where esigma is A051377.

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A346533 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000203 together with the sum of A000203(n-1) and A000203(n), with a(1) = 1.

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A350106 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j * floor(n/j)^k.

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A365409 a(n) = Sum_{k=1..n} binomial(floor(n/k)+3,4).

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A024917 a(n) = Sum_{k=2..n} k*floor(n/k).

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Magma

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A086718 Convolution of sequence of primes with sequence sigma(n).

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Maple

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A123327 a(n) = A000203(n) + A004125(n).

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A134867 A010766 * A000012.

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