A004125 -id:A004125 - cp's OEIS frontend

A237590 a(n) is the total number of regions (or parts) after n-th stage in the diagram of the symmetries of sigma described in A236104.

1, 2, 4, 5, 7, 8, 10, 11, 14, 16, 18, 19, 21, 23, 26, 27, 29, 30, 32, 33, 37, 39, 41, 42, 45, 47, 51, 52, 54, 55, 57, 58, 62, 64, 67, 68, 70, 72, 76, 77, 79, 80, 82, 84, 87, 89, 91, 92, 95, 98, 102, 104, 106, 107, 111, 112, 116, 118, 120, 121, 123, 125, 130, 131, 135, 136, 138, 140, 144, 147, 149, 150, 152, 154

Offset: 1

Omar E. Pol, Mar 31 2014

nonn

The total area (or total number of cells) of the diagram after n stages is equal to A024916(n), the sum of all divisors of all positive integers <= n.
Note that the region between the virtual circumscribed square and the diagram is a symmetric polygon whose area is equal to A004125(n), see example.
For more information see A237593 and A237270.
a(n) is also the total number of terraces of the stepped pyramid with n levels described in A245092. - Omar E. Pol, Apr 20 2016

			Illustration of initial terms:
.                                                         _ _ _ _
.                                           _ _ _        |_ _ _  |_
.                               _ _ _      |_ _ _|       |_ _ _|   |_
.                     _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.             _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.       _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.      |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.
.       1      2        4          5            7              8
.
For n = 6 the diagram contains 8 regions (or parts), so a(6) = 8.
The sum of all divisors of all positive integers <= 6 is [1] + [1+2] + [1+3] + [1+2+4] + [1+5] + [1+2+3+6] = 33. On the other hand after 6 stages the sum of all parts of the diagram is [1] + [3] + [2+2] + [7] + [3+3] + [12] = 33, equaling the sum of all divisors of all positive integers <= 6.
Note that the region between the virtual circumscribed square and the diagram is a symmetric polygon whose area is equal to A004125(6) = 3.
From _Omar E. Pol_, Dec 25 2020: (Start)
Illustration of the diagram after 29 stages (contain 215 vertices, 268 edges and 54 regions or parts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
|_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
|_ _ _ _ _ _ _ _ _ _ _ _ _  | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
|_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
|_ _ _ _ _ _ _ _ _ _  | |       |_ _|   |_
|_ _ _ _ _ _ _ _ _ _| | |_ _    |_  |_ _  |_ _
|_ _ _ _ _ _ _ _ _  | |_ _ _|     |_  | |_ _  |
|_ _ _ _ _ _ _ _ _| | |_ _  |_      |_|_ _  | |
|_ _ _ _ _ _ _ _  | |_ _  |_ _|_        | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| |     |     | |_ _    | |_|_ _ _ _ _  | |
|_ _ _ _ _ _ _  | |_ _  |_    |_  | |   |_ _ _ _ _  | | | |
|_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
|_ _ _ _ _ _  | |_  |_  |_    | |_|_ _ _ _  | | | | | | | |
|_ _ _ _ _ _| |_ _|   |_  |   |_ _ _ _  | | | | | | | | | |
|_ _ _ _ _  |     |_ _  | |_ _ _ _  | | | | | | | | | | | |
|_ _ _ _ _| |_      | |_|_ _ _  | | | | | | | | | | | | | |
|_ _ _ _  |_ _|_    |_ _ _  | | | | | | | | | | | | | | | |
|_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
|_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
|_ _ _|   |_ _  | | | | | | | | | | | | | | | | | | | | | |
|_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
|_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
(End)

Robert Price, Table of n, a(n) for n = 1..5000
Omar E. Pol, An infinite stepped pyramid
Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the stepped pyramid (first 16 levels)

Partial sums of A237271.
Compare with A060831 (analog for the diagram that contains subparts).
Cf. A000203, A004125, A024916, A196020, A236104, A235791, A237048, A237270, A237591, A237593, A239659, A239660, A239663, A239665, A239931-A239934, A245092, A244050, A244970, A262626, A317109.

(* total number of parts in the first n symmetric representations *)
(* Function a237270[] is defined in A237270 *)
(* variable "previous" represents the sum from 1 through m-1 *)
a237590[previous_,{m_,n_}]:=Rest[FoldList[Plus[#1,Length[a237270[#2]]]&,previous,Range[m,n]]]
a237590[n_]:=a237590[0,{1,n}]
a237590[78] (* data *)
(* Hartmut F. W. Hoft, Jul 07 2014 *)

a(n) = A317109(n) - A294723(n) + 1 (Euler's formula). - Omar E. Pol, Jul 21 2018

Definition clarified by Omar E. Pol, Jul 21 2018

A024934 Sum of remainders n mod p, over all primes p < n.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 1, 4, 6, 7, 4, 8, 8, 13, 10, 8, 12, 18, 20, 27, 28, 26, 21, 29, 33, 37, 31, 37, 37, 46, 46, 56, 65, 62, 54, 53, 59, 70, 61, 57, 62, 74, 75, 88, 89, 95, 84, 98, 108, 116, 124, 119, 119, 134, 145, 145, 152, 146, 131, 147, 154, 171, 156, 164, 180, 180, 182, 200, 200, 193, 198, 217

Offset: 0

Clark Kimberling

nonn

			a(5) = 3. The remainder when 5 is divided by primes 2, 3 respectively is 1, 2, and their sum = 3.
10 = 2*5+0 = 3*3+1 = 5*2+0 = 7*1+3: a(10) = 0+1+0+3 = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A004125, A067435, A067436, A013939, A101336.

a[n_] := Sum[Mod[n, Prime[i]], {i, PrimePi@ n}]; Array[a, 72, 0] (* Giovanni Resta, Jun 24 2016 *)
Table[Total[Mod[n,Prime[Range[PrimePi[n]]]]],{n,0,80}] (* Harvey P. Dale, Jul 02 2025 *)

a(n)=my(r=0);forprime(p=2,n,r+=n%p); r; \\ Joerg Arndt, Nov 05 2016

a(n) = n*A000720(n) - A024924(n). - Max Alekseyev, Feb 10 2012

a(n) = a(n-1) + A000720(n-1) - A105221(n). - Max Alekseyev, Nov 28 2017

Edited by Max Alekseyev, Jan 30 2012

a(0)=0 prepended by Max Alekseyev, Dec 10 2013

A236112 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists k+1 copies of the squares in nondecreasing order, and the first element of column k is in row k(k+1)/2.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 4, 0, 4, 1, 0, 9, 1, 0, 9, 1, 0, 16, 4, 0, 16, 4, 1, 0, 25, 4, 1, 0, 25, 9, 1, 0, 36, 9, 1, 0, 36, 9, 4, 0, 49, 16, 4, 1, 0, 49, 16, 4, 1, 0, 64, 16, 4, 1, 0, 64, 25, 9, 1, 0, 81, 25, 9, 1, 0, 81, 25, 9, 4, 0, 100, 36, 9, 4, 1, 0, 100, 36, 16, 4, 1, 0, 121, 36, 16, 4, 1, 0, 121, 49, 16, 4, 1, 0

Offset: 1

Omar E. Pol, Jan 23 2014

Gives an identity for the sum of remainders of n mod k, for k = 1,2,3,...,n. Alternating sum of row n equals A004125(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A004125(n).

Row n has length A003056(n) hence the first element of column k is in row A000217(k).

			Triangle begins:
    0;
    0;
    1,   0;
    1,   0;
    4,   0;
    4,   1,   0;
    9,   1,   0;
    9,   1,   0;
   16,   4,   0;
   16,   4,   1,   0;
   25,   4,   1,   0;
   25,   9,   1,   0;
   36,   9,   1,   0;
   36,   9,   4,   0;
   49,  16,   4,   1,  0;
   49,  16,   4,   1,  0;
   64,  16,   4,   1,  0;
   64,  25,   9,   1,  0;
   81,  25,   9,   1,  0;
   81,  25,   9,   4,  0;
  100,  36,   9,   4,  1,  0;
  100,  36,  16,   4,  1,  0;
  121,  36,  16,   4,  1,  0;
  121,  49,  16,   4,  1,  0;
  ...
For n = 24 the 24th row of triangle is 121, 49, 16, 4, 1, 0 therefore the alternating row sum is 121 - 49 + 16 - 4 + 1 - 0 = 85 equaling A004125(24).

Cf. A000203, A000217, A000290, A003056, A004125, A120444, A196020, A211343, A228813, A231345, A231347, A235791, A235794, A236104, A236106, A237048, A237591, A237593, A261699.

A244049 Sum of all proper divisors of all positive integers <= n.

Original entry on oeis.org

0, 0, 0, 2, 2, 7, 7, 13, 16, 23, 23, 38, 38, 47, 55, 69, 69, 89, 89, 110, 120, 133, 133, 168, 173, 188, 200, 227, 227, 268, 268, 298, 312, 331, 343, 397, 397, 418, 434, 483, 483, 536, 536, 575, 607, 632, 632, 707, 714, 756, 776, 821, 821, 886, 902

Offset: 1

Omar E. Pol, Jun 24 2014

The proper divisors of n are all divisors except 1 and n itself. Therefore noncomposite numbers have no proper divisors.
For the sum of all aliquot divisors of all positive integers <= n see A153485.
For the sum all divisors of all positive integers <= n see A024916.
a(n) = a(n - 1) if and only if n is prime.
For n >= 3 a(n) equals the area of an arrowhead-shaped polygon formed by two zig-zag paths and the Dyck path described in the n-th row of A237593 as shown in the Links section. Note that there is a similar diagram of A153485(n) in A153485. - Omar E. Pol, Jun 14 2022

			a(4) = 2 because the only proper divisor of 4 is 2 and the previous n contributed no proper divisors to the sum.
a(5) = 2 because 5 is prime and contributes no proper divisors to the sum.
a(6) = 7 because the proper divisors of 6 are 2 and 3, which add up to 5, and a(5) + 5 = 2 + 5 = 7.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Omar E. Pol, Illustration of initial terms with arrowhead-shaped polygons.

Partial sums of A048050.

Cf. A000040, A000203, A000290, A001065, A004125, A008578, A024916, A027750, A034856, A153485, A161680, A237591, A237593, A245092, A262626.

propDivsRunSum[1] := 0; propDivsRunSum[n_] := propDivsRunSum[n] = propDivsRunSum[n - 1] + (Plus@@Divisors[n]) - (n + 1); Table[propDivsRunSum[n], {n, 60}] (* Alonso del Arte, Jun 30 2014 *)
Accumulate[Join[{0},Table[Total[Most[Divisors[n]]]-1,{n,2,60}]]] (* Harvey P. Dale, Aug 12 2016 *)
Accumulate[Join[{0}, Table[DivisorSigma[1, n] - n - 1, {n, 2, 55}]]] (* Amiram Eldar, Jun 18 2022 *)

a(n) = sum(k=2, n, sigma(k)-k-1); \\ Michel Marcus, Mar 30 2021

from math import isqrt
def A244049(n): return ((-n*(n+3)-(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 21 2023

a(n) = A024916(n) - A034856(n).
a(n) = A153485(n) - n + 1.
G.f.: (1/(1 - x))*Sum_{k>=2} k*x^(2*k)/(1 - x^k). - Ilya Gutkovskiy, Jan 22 2017
a(n) = A161680(n-1) - A004125(n). - Omar E. Pol, Mar 25 2021
a(n) = A000290(n) - A034856(n) - A004125(n). - Omar E. Pol, Mar 26 2021
a(n) = c * n^2 + O(n*log(n)), where c = Pi^2/12 - 1/2 = 0.322467... . - Amiram Eldar, Nov 27 2023

A235796 2*n - 1 - sigma(n).

Original entry on oeis.org

0, 0, 1, 0, 3, -1, 5, 0, 4, 1, 9, -5, 11, 3, 5, 0, 15, -4, 17, -3, 9, 7, 21, -13, 18, 9, 13, -1, 27, -13, 29, 0, 17, 13, 21, -20, 35, 15, 21, -11, 39, -13, 41, 3, 11, 19, 45, -29, 40, 6, 29, 5, 51, -13, 37, -9, 33, 25, 57, -49, 59, 27, 21, 0, 45, -13, 65, 9, 41

Offset: 1

Omar E. Pol, Jan 25 2014

sign

Partial sums give A004125.
Also 0 together with A120444.
It appears that a(n) = 0 iff n is a power of 2.
Numbers n with a(n) = 0 are called "almost perfect", "least deficient" or "slightly defective" numbers.  See A000079. - Robert Israel, Jul 22 2014
a(n) = n - 2 iff n is prime.
a(n) = -1 iff n is a perfect number.
Also the alternating row sums of A239446. - Omar E. Pol, Jul 21 2014

			.     The positive     The sum of
n     odd numbers     divisors of n.      a(n)
1          1                1               0
2          3                3               0
3          5                4               1
4          7                7               0
5          9                6               3
6         11               12              -1
7         13                8               5
8         15               15               0
9         17               13               4
10        19               18               1
...

R. K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, New York, 2004.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000079, A000203, A000396, A004125, A005408, A120444, A196020, A236104.

[2*n-1-SumOfDivisors(n): n in [1..100]]; // Vincenzo Librandi, Feb 25 2014

Table[2n-1-DivisorSigma[1,n],{n,70}] (* Harvey P. Dale, Jul 11 2014 *)

vector(100, n, (2*n-1)-sigma(n)) \\ Colin Barker, Jan 27 2014

a(n) = A005408(n-1) - A000203(n).
a(n) = -1 - A033880(n). - Michel Marcus, Jan 27 2014
a(n) =  n - 1 - A001065(n). - Omar E. Pol, Jan 29 2014
a(n) = A033879(n) - 1. - Omar E. Pol, Jan 30 2014
a(n) = 2*n - 2 - A039653(n). - Omar E. Pol, Jan 31 2014
a(n) = (-1)*A237588(n). - Omar E. Pol, Feb 23 2014
a(n) = 2*n - A088580(n). - Omar E. Pol, Mar 23 2014

A051127 Table T(n,k) = k mod n read by antidiagonals (n >= 1, k >= 1).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Note that the upper right half of this sequence when formatted as a square array is essentially the same as this whole sequence when formatted as an upper right triangle. Sums of antidiagonals are A004125. - Henry Bottomley, Jun 22 2001

			0  0  0  0  0  0  0  0  0  0 ...
1  0  1  0  1  0  1  0  1  0 ...
1  2  0  1  2  0  1  2  0  1 ...
1  2  3  0  1  2  3  0  1  2 ...
1  2  3  4  0  1  2  3  4  0 ...
1  2  3  4  5  0  1  2  3  4 ...
1  2  3  4  5  6  0  1  2  3 ...
1  2  3  4  5  6  7  0  1  2 ...
1  2  3  4  5  6  7  8  0  1 ...
1  2  3  4  5  6  7  8  9  0 ...
1  2  3  4  5  6  7  8  9 10 ...
1  2  3  4  5  6  7  8  9 10 ...
1  2  3  4  5  6  7  8  9 10 ...

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Transpose of A051126.
Cf. A048158, A051777, A122750.
Cf. A002260, A004125, A004736.

T[n_, m_] = Mod[n - m + 1, m + 1]; Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%] (* Roger L. Bagula, Sep 04 2008 *)

T(n, k)=k%n \\ Charles R Greathouse IV, Feb 09 2017

As a linear array, the sequence is a(n) = A004736(n) mod A002260(n) or a(n) = ((t*t+3*t+4)/2-n) mod (n-(t*(t+1)/2)), where t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 17 2012

G.f. for the n-th row: y*Sum_{i=0..n-2} (i + 1)*y^i/(1 - y^n). - Stefano Spezia, May 08 2024

More terms from James Sellers, Dec 11 1999

A103288 Numbers k such that sigma(k) >= 2k-1 (union of perfect, abundant and least deficient numbers).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 128, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220

Offset: 1

Max Alekseyev, Jan 28 2005

nonn

If the only least deficient numbers are the powers of 2 (open problem) then this sequence is the union of A023196 and A000079.

Like the abundant numbers, this sequence has density between 0.2474 and 0.2480, see A005101. - Charles R Greathouse IV, Nov 30 2022

Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A023196, A023197, A023198, A023199, A000079, A004125.

Select[Range[500], DivisorSigma[1, #] >= 2*# - 1 &] (* Paolo Xausa, Dec 09 2024 *)

for(n=1,1000,if(sigma(n)>=2*n-1,print(n)));

Numbers k such that A004125(k) <= A004125(k-1).

A235794 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k starts with k zeros and then lists the odd numbers interleaved with k zeros, and the first element of column k is in row k(k+1)/2.

Original entry on oeis.org

0, 1, 0, 0, 3, 0, 0, 1, 5, 0, 0, 0, 0, 0, 7, 3, 0, 0, 0, 1, 9, 0, 0, 0, 0, 5, 0, 0, 11, 0, 0, 0, 0, 0, 3, 0, 13, 7, 0, 1, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 9, 5, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 3, 0, 19, 11, 0, 0, 1, 0, 0, 7, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0

Offset: 1

Omar E. Pol, Jan 23 2014

It appears that the alternating row sums give A120444, the first differences of A004125, i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A120444(n).

Row n has length A003056(n) hence the first element of column k is in row A000217(k).

			Triangle begins:
  0;
  1;
  0,  0;
  3,  0;
  0,  1;
  5,  0,  0;
  0,  0,  0;
  7,  3,  0;
  0,  0,  1;
  9,  0,  0,  0;
  0,  5,  0,  0;
  11, 0,  0,  0;
  0,  0,  3,  0;
  13, 7,  0,  1;
  0,  0,  0,  0,  0;
  15, 0,  0,  0,  0;
  0,  9,  5,  0,  0;
  17, 0,  0,  0,  0;
  0,  0,  0,  3,  0;
  19, 11, 0,  0,  1;
  0,  0,  7,  0,  0,  0;
  21, 0,  0,  0,  0,  0;
  0,  13, 0,  0,  0,  0;
  23, 0,  0,  5,  0,  0;
  ...
For n = 14 the 14th row of triangle is 13, 7, 0, 1, and the alternating sum is 13 - 7 + 0 - 1 = 5, the same as A120444(14) = 5.

Cf. A000203, A000217, A003056, A004125, A120444, A196020, A211343, A228813, A231345, A231347, A235791, A236104, A236106, A236112.

A243980 Four times the sum of all divisors of all positive integers <= n.

Original entry on oeis.org

4, 16, 32, 60, 84, 132, 164, 224, 276, 348, 396, 508, 564, 660, 756, 880, 952, 1108, 1188, 1356, 1484, 1628, 1724, 1964, 2088, 2256, 2416, 2640, 2760, 3048, 3176, 3428, 3620, 3836, 4028, 4392, 4544, 4784, 5008, 5368, 5536, 5920, 6096, 6432, 6744, 7032, 7224, 7720

Offset: 1

Omar E. Pol, Jun 18 2014

nonn

Also number of "ON" cells at n-th stage in a structure which looks like a simple 2-dimensional cellular automaton (see example). The structure is formed by the reflection on the four quadrants from the diagram of the symmetry of sigma in the first quadrant after n-th stage, hence the area in each quadrant equals the area of each wedge and equals A024916(n); the sum of all divisors of all positive integers <= n. For more information about the diagram see A237593 and A237270.

			Illustration of the structure after 16 stages (contains 880 ON cells):
.                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
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.

Indranil Ghosh, Table of n, a(n) for n = 1..7342

Partial sums of A239050.
Partial sums give A244050.
Cf. A000203, A000290, A004125, A016742, A024916, A175254, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934.

Accumulate[4*DivisorSigma[1,Range[50]]] (* Harvey P. Dale, May 13 2018 *)

from math import isqrt
def A243980(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

a(n) = A016742(n) - 4*A004125(n) = 4*A024916(n).

a(n) = 2*(A006218(n) + A222548(n)) = 2*A327329(n). - Omar E. Pol, Sep 25 2019

A067436 a(n) = sum of all the remainders when n-th even number is divided by even numbers < 2n.

Original entry on oeis.org

0, 0, 2, 2, 8, 6, 16, 16, 24, 26, 44, 34, 56, 62, 72, 72, 102, 94, 128, 122, 140, 154, 196, 170, 206, 224, 250, 248, 302, 276, 334, 334, 368, 394, 436, 396, 466, 496, 538, 516, 594, 568, 650, 656, 678, 716, 806, 748, 828, 840, 898, 908, 1010, 984, 1058, 1040

Offset: 1

Amarnath Murthy, Jan 29 2002

			a(5) = 8. The remainder when 10 is divided by 4,6,8, respectively is 2,4,2 and their sum = 8.

Cf. A013661, A004125, A067435.

Accumulate[Table[4*n - 2*DivisorSigma[1, n] - 2, {n, 1, 100}]] (* Amiram Eldar, Mar 30 2024 *)

from math import isqrt
def A067436(n): return (n**2<<1)+(s:=isqrt(n))**2*(s+1)-sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1)) # Chai Wah Wu, Oct 22 2023

a(n) = 2*A004125(n).

a(n) = (2 - Pi^2/6) * n^2 + O(n*log(n)). - Amiram Eldar, Mar 30 2024

Corrected and extended by several contributors.

cp's OEIS Frontend

A237590 a(n) is the total number of regions (or parts) after n-th stage in the diagram of the symmetries of sigma described in A236104.

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A024934 Sum of remainders n mod p, over all primes p < n.

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A236112 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists k+1 copies of the squares in nondecreasing order, and the first element of column k is in row k(k+1)/2.

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A244049 Sum of all proper divisors of all positive integers <= n.

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A235796 2*n - 1 - sigma(n).

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Magma

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A051127 Table T(n,k) = k mod n read by antidiagonals (n >= 1, k >= 1).

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A103288 Numbers k such that sigma(k) >= 2k-1 (union of perfect, abundant and least deficient numbers).

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