A153485 -id:A153485 - cp's OEIS frontend

A279847 a(n) = Sum_{k=1..n} k^2*(floor(n/k) - 1).

0, 1, 2, 7, 8, 22, 23, 44, 54, 84, 85, 151, 152, 206, 241, 326, 327, 458, 459, 605, 664, 790, 791, 1065, 1091, 1265, 1356, 1622, 1623, 2023, 2024, 2365, 2496, 2790, 2865, 3480, 3481, 3847, 4026, 4636, 4637, 5373, 5374, 6000, 6341, 6875, 6876, 7982, 8032, 8787, 9086, 9952, 9953, 11137, 11284

Offset: 1

Ilya Gutkovskiy, Dec 20 2016

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

Total volume of all rectangular prisms with dimensions (x, x, z) and integers x and y, such that x + y = n, 0 < x <= y, and z = floor(y/x). - Wesley Ivan Hurt, Dec 21 2020

			For n = 7 the proper divisors of the first seven positive integers are {0}, {1}, {1}, {1, 2}, {1}, {1, 2, 3}, {1} so a(7) = 0^2 + 1^2 + 1^2 + 1^2 + 2^2 + 1 ^2 + 1^2 + 2^2 + 3^2 + 1^2 = 23.

Index entries for sequences related to sums of divisors

Cf. A000290, A000330, A001157, A064602, A067558, A153485.

Table[Sum[k^2 (Floor[n/k] - 1), {k, 1, n}], {n, 55}]
Table[Sum[DivisorSigma[2, k] - k^2, {k, 1, n}], {n, 55}]

a(n) = sum(k=1, n, k^2*(floor(n/k)-1)) \\ Felix Fröhlich, Dec 20 2016

from math import isqrt
def A279847(n): return (-n*(n+1)*(2*n+1)-(s:=isqrt(n))**2*(s+1)*(2*s+1) + sum((q:=n//k)*(6*k**2+q*(2*q+3)+1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 21 2023

G.f.: -x*(1 + x)/(1 - x)^4 + (1/(1 - x))*Sum_{k>=1} k^2*x^k/(1 - x^k).
a(n) = A064602(n) - A000330(n).
a(n) = Sum_{k=1..n} A067558(k).
a(n) = Sum_{k=1..n} (A001157(k) - A000290(k)).
a(p^k) = a(p^k-1) + (p^(2*k) - 1)/(p^2 - 1), when p is prime.
a(n) ~ ((zeta(3) - 1)/3)*n^3.
a(n) = Sum_{k=1..floor(n/2)} k^2 * floor((n-k)/k). - Wesley Ivan Hurt, Dec 21 2020

A299692 a(n) is the total area that is visible in the perspective view of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

3, 10, 20, 35, 51, 75, 97, 128, 159, 197, 231, 283, 323, 375, 429, 492, 544, 619, 677, 759, 833, 913, 983, 1091, 1172, 1266, 1360, 1472, 1560, 1692, 1786, 1913, 2027, 2149, 2267, 2430, 2542, 2678, 2812, 2982, 3106, 3286, 3416, 3588, 3756, 3920, 4062, 4282, 4437, 4630, 4804, 5006, 5166, 5394, 5576, 5808, 6002

Offset: 1

Omar E. Pol, Mar 06 2018

nonn

a(n) is also the sum of all divisors of all positive integers <= n, plus the n-th oblong number, since A024916(n) equals the total area of the horizontal terraces of the stepped pyramid with n levels, and A002378(n) equals the total area of the vertical sides that are visible (see link).

a(n) is also the sum of all aliquot divisors of all positive integers <= n, plus the n-th triangular matchstick number.

			For n = 3 the areas of the terraces of the first three levels starting from the top of the stepped pyramid are 1, 3 and 4 respectively. On the other hand the areas of the vertical sides that are visible are [1, 1], [2, 2], [2, 1, 1, 2], or in successive levels 2, 4, 6 respectively. Hence the total area that is visible is equal to 1 + 3 + 4 + 2 + 4 + 6 = 8 + 12 = 20, so a(3) = 20.
For n = 16 the total number of horizontal and vertical cells that are visible are 220 and 272 respectively. So a(16) = 220 + 272 = 492 (see the link).

Omar E. Pol, Perspective view of the pyramid with 16 levels, contains 492 cells.

Partial sums of A224880.

Cf. A002378, A024916, A045943, A072691, A153485, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A244050, A245092, A262626, A328366.

Accumulate[Table[DivisorSigma[1, n] + 2*n, {n, 1, 50}]] (* Amiram Eldar, Mar 21 2024 *)

a(n) = sum(k=1, n, n\k*k) + n*(n+1); \\ Michel Marcus, Jun 21 2018

from math import isqrt
def A299692(n): return n*(n+1)+(-(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 22 2023

a(n) = A024916(n) + A002378(n).
a(n) = A153485(n) + A045943(n).
a(n) = A328366(n)/2. - Omar E. Pol, Apr 22 2020
a(n) = c * n^2 + O(n*log(n)), where c = zeta(2)/2 + 1 = A072691 + 1 = 1.822467... . - Amiram Eldar, Mar 21 2024

A235799 a(n) = n^2 - sigma(n).

Original entry on oeis.org

0, 1, 5, 9, 19, 24, 41, 49, 68, 82, 109, 116, 155, 172, 201, 225, 271, 285, 341, 358, 409, 448, 505, 516, 594, 634, 689, 728, 811, 828, 929, 961, 1041, 1102, 1177, 1205, 1331, 1384, 1465, 1510, 1639, 1668, 1805, 1852, 1947, 2044, 2161, 2180, 2344, 2407

Offset: 1

Omar E. Pol, Jan 24 2014

nonn

From Omar E. Pol, Apr 11 2021: (Start)
If n is prime (A000040) then a(n) = n^2 - n - 1.
If n is a power of 2 (A000079) then a(n) = (n-1)^2.
If n is a perfect number (A000396) then a(n) = (n-1)^2 - 1, assuming there are no odd perfect numbers.
In order to construct the diagram of the symmetric representation of a(n) we use the following rules:
At stage 1 in the first quadrant of the square grid we draw the symmetric representation of sigma(n) using the two Dyck paths described in the rows n and n-1 of A237593. The area of the region that is below the symmetric representation of sigma(n) equals A024916(n-1).
At stage 2 we draw a pair of orthogonal line segments (if it's necessary) such that in the drawing appears totally formed a square n X n. The area of the region that is above the symmetric representation of sigma(n) equals A004125(n).
At stage 3 we turn OFF the cells of the symmetric representation of sigma(n). Then we turn ON the rest of the cells that are in the square n X n. The result is that the ON cell form the diagram of the symmetric representation of a(n). See the Example section. (End)

			From _Omar E. Pol, Apr 04 2021: (Start)
Illustration of initial terms in the first quadrant for n = 1..6:
.
.                                                             y|        _ _
.                                              y|      _ _     |_ _ _  |_  |
.                                 y|      _     |_ _ _|   |    |     |   |_|
.                      y|    _     |_ _  |_|    |        _|    |     |_ _
.             y|        |_ _|_|    |   |_       |       |      |         |
.      y|      |_       |   |      |     |      |       |      |         |
.       |_ _   |_|_ _   |_ _|_ _   |_ _ _|_ _   |_ _ _ _|_ _   |_ _ _ _ _|_ _
.          x        x          x            x              x                x
.
n:        1       2         3           4             5               6
a(n):     0       1         5           9            19              24
.
Illustration of initial terms in the first quadrant for n = 7..9:
.                                                y|          _ _ _ _
.                          y|          _ _ _      |_ _ _ _ _|       |
.      y|        _ _ _      |_ _ _ _  |     |     |          _ _    |
.       |_ _ _ _|     |     |       | |_    |     |         |_  |   |
.       |             |     |       |_  |_ _|     |           |_|  _|
.       |            _|     |         |_ _        |               |
.       |           |       |             |       |               |
.       |           |       |             |       |               |
.       |           |       |             |       |               |
.       |_ _ _ _ _ _|_ _    |_ _ _ _ _ _ _|_ _    |_ _ _ _ _ _ _ _|_ _
.                      x                     x                       x
.
n:              7                    8                      9
a(n):          41                   49                     68
.
For n = 9 the figures 1, 2 and 3 below show respectively the three stages described in the Comments section as follows:
.
.   y|_ _ _ _ _ 5            y|_ _ _ _ _ _ _ _ _      y|          _ _ _ _
.    |_ _ _ _ _|              |_ _ _ _ _|       |      |_ _ _ _ _|       |
.    |         |_ _ 3         |         |_ _ R  |      |          _ _    |
.    |         |_  |          |         |_  |   |      |         |_  |   |
.    |           |_|_ _ 5     |           |_|_ _|      |           |_|  _|
.    |               | |      |               | |      |               |
.    |      Q        | |      |       Q       | |      |               |
.    |               | |      |               | |      |               |
.    |               | |      |               | |      |               |
.    |_ _ _ _ _ _ _ _|_|_     |_ _ _ _ _ _ _ _|_|_     |_ _ _ _ _ _ _ _|_ _
.                       x                        x                        x
.         Figure 1.                Figure 2.                Figure 3.
.         Symmetric                Symmetric                Symmetric
.       representation           representation           representation
.         of sigma(9)              of sigma(9)             of a(9) = 68
.       A000203(9) = 13          A000203(9) = 13
.           and of                   and of
.     Q = A024916(8) = 56      R = A004125(9) = 12
.                              Q = A024916(8) = 56
.
Note that the symmetric representation of a(9) contains a hole formed by three cells because these three cells were the central part of the symmetric representation of sigma(9). (End)

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

Cf. A000040, A000079, A000203, A000217, A000290, A000396, A004125, A024816, A024916, A067436, A120444, A153485, A196020, A236104, A236112, A237593, A244048, A342344.

[n^2 - DivisorSigma(1,n): n in [1..50]]; // G. C. Greubel, Oct 31 2018

Table[n^2-DivisorSigma[1,n],{n,50}] (* Harvey P. Dale, Sep 02 2016 *)

vector(50, n, n^2 - sigma(n)) \\ G. C. Greubel, Oct 31 2018

a(n) = A000290(n) - A000203(n).
a(n) = A024916(n-1) + A004125(n), n > 1.
G.f.: x*(1 + x)/(1 - x)^3 - Sum_{k>=1} x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Mar 17 2017
From Omar E. Pol, Apr 10 2021: (Start)
a(n) = A024816(n) + A000217(n-1).
a(n) = A067436(n) + A153485(n) + A244048(n). (End)

A236540 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists k copies of the positive squares in nondecreasing order, except the first column which lists the triangular numbers, and the first element of column k is in row k(k+1)/2.

Original entry on oeis.org

0, 1, 3, 1, 6, 1, 10, 4, 15, 4, 1, 21, 9, 1, 28, 9, 1, 36, 16, 4, 45, 16, 4, 1, 55, 25, 4, 1, 66, 25, 9, 1, 78, 36, 9, 1, 91, 36, 9, 4, 105, 49, 16, 4, 1, 120, 49, 16, 4, 1, 136, 64, 16, 4, 1, 153, 64, 25, 9, 1, 171, 81, 25, 9, 1, 190, 81, 25, 9, 4, 210, 100, 36, 9, 4, 1

Offset: 1

Omar E. Pol, Jan 28 2014

Gives an identity for the sum of all aliquot divisors of all positive integers <= n.
Alternating sum of row n equals A153485(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A153485(n).
Row n has length A003056(n) hence the first element of column k is in row A000217(k).
Column 1 is A000217. Columns 2-3 are A008794, A211547, but without the zeros.
Column k lists the partial sums of the k-th column of triangle A231347 which gives an identity for the sum of aliquot divisors of n. - Omar E. Pol, Nov 11 2014

			Triangle begins:
    0;
    1;
    3,   1;
    6,   1;
   10,   4;
   15,   4,   1;
   21,   9,   1;
   28,   9,   1;
   36,  16,   4;
   45,  16,   4,   1;
   55,  25,   4,   1;
   66,  25,   9,   1;
   78,  36,   9,   1;
   91,  36,   9,   4;
  105,  49,  16,   4,  1;
  120,  49,  16,   4,  1;
  136,  64,  16,   4,  1;
  153,  64,  25,   9,  1;
  171,  81,  25,   9,  1;
  190,  81,  25,   9,  4;
  210, 100,  36,   9,  4,  1;
  231, 100,  36,  16,  4,  1;
  253, 121,  36,  16,  4,  1;
  276, 121,  49,  16,  4,  1;
  ...
For n = 6 the divisors of all positive integers <= 6 are [1], [1, 2], [1, 3], [1, 2, 4], [1, 5], [1, 2, 3, 6] hence the sum of all aliquot divisors is [0] + [1] + [1] + [1+2] + [1] + [1+2+3] = 0 + 1 + 1 + 3 + 1 + 6 = 12. On the other hand the 6th row of triangle is 15, 4, 1, therefore the alternating row sum is 15 - 4 + 1 = 12, equaling the sum of all aliquot divisors of all positive integers <= 6.

Cf. A000203, A000217, A001065, A008794, A003056, A153485, A196020, A211547, A211343, A228813, A231345, A231347, A235791, A235794, A235799, A236104, A236106, A236112, A237591, A237593, A286001.

A284755 Numbers n such that the average of all proper divisors of all positive integers <= n is an integer.

Original entry on oeis.org

2, 3, 63, 1249, 4696, 1200509

Offset: 1

Ilya Gutkovskiy, Apr 02 2017

Numbers n such that A002541(n)|A153485(n).

a(7) > 10^12. - Giovanni Resta, Apr 13 2017

Eric Weisstein's World of Mathematics, Restricted Divisor Function
Index entries for sequences related to sums of divisors

Cf. A000005, A000203, A001065, A002541, A003601, A023884, A023886, A027751, A153485, A226647, A284288.

Select[Range[2, 1300000], Mod[Sum[DivisorSigma[1, k] - k, {k, 1, #}], Sum[DivisorSigma[0, k] - 1, {k, 1, #}]] == 0 &]

A339370 a(n) = Sum_{k=1..floor(n/2)} (n-k) * floor((n-k)/k).

Original entry on oeis.org

0, 1, 4, 11, 19, 36, 50, 76, 102, 138, 165, 227, 262, 318, 381, 460, 510, 614, 672, 791, 889, 990, 1064, 1249, 1353, 1477, 1610, 1790, 1891, 2133, 2244, 2455, 2626, 2798, 2983, 3312, 3452, 3649, 3857, 4198, 4356, 4715, 4883, 5190, 5514, 5763, 5949, 6446, 6686, 7045

Offset: 1

Wesley Ivan Hurt, Dec 01 2020

Total area of all y X z rectangles such that x + y = n, with x and y integers, 0 < x <= y and z = floor(y/x). - Wesley Ivan Hurt, Dec 21 2020

Cf. A002541, A153485.

Table[Sum[(n - k)*Floor[(n - k)/k], {k, Floor[n/2]}], {n, 60}]

a(n) = sum(k=1, n\2, (n-k) * ((n-k)\k)); \\ Michel Marcus, Dec 02 2020

from math import isqrt
def A339370(n): return n*(1-n)+(s:=isqrt(n))**2*(s+1-(n<<1))-sum((q:=n//k)*((k-(n<<1)<<1)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 27 2023

a(n) ~ n^2*(log(n) + 2*EulerGamma - Pi^2/12 - 3/2). - Rok Cestnik, Dec 20 2020

a(n) = n*A002541(n) - A153485(n). - Vaclav Kotesovec, Dec 21 2020

A342344 Number of parts in the symmetric representation of antisigma(n).

Original entry on oeis.org

0, 0, 2, 3, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2

Offset: 1

Omar E. Pol, Mar 08 2021

nonn

In order to construct this sequence and the diagram of the symmetric representation of antisigma(n) = A024816(n) we use the following rules:
At stage 1 in the first quadrant of the square grid we draw the symmetric representation of sigma(n) using the two Dyck paths described in the rows n and n-1 of A237593. The area of the region that is below the symmetric representation of sigma(n) equals A024916(n-1).
At stage 2 we draw a pair of orthogonal line segments (if it's necessary) such that in the drawing appears totally formed a square n X n. The area of the region that is above the symmetric representation of sigma(n) equals A004125(n). Then we draw a zig-zag path with line segments of length 1 from (0,n-1) to (n-1,0) such that appears a staircase with n-1 steps. The area of the region (or regions) that is below the symmetric representation of sigma(n) and above the staircase equals A244048(n) = A153485(n-1). The area of the region that is below the staircase equals A000217(n-1).
At stage 3 we turn OFF the cells of the symmetric representation of sigma(n) and also the cells that are below the staircase. Then we turn ON the rest of the cells that are in the square n X n. The result is that the ON cell form the diagram of the symmetric representation of antisigma(n) = A024816(n). See the Example section.
For n >= 7; if A237271(n) = 1 or n is a term of A262259 then a(n) = 2 otherwise a(n) = 1.

			Illustration of the symmetric representation of antisigma(n) = AS(n) = A024816(n), for n = 1..6:
.                                                             y|        _ _
.                                              y|      _ _     |  _ _  |_  |
.                                 y|      _     |  _ _|   |    | |_  |   |_|
.                      y|    _     |  _  |_|    | |_     _|    |   |_|_ _
.             y|        |  _|_|    | |_|_       |   |_  |      |     |_  |
.      y|      |        | |_|      |   |_|      |     |_|      |       |_|
.       |_ _   |_ _ _   |_ _ _ _   |_ _ _ _ _   |_ _ _ _ _ _   |_ _ _ _ _ _ _
.          x        x          x            x              x                x
.
n:        1       2         3           4             5               6
a(n):     0       0         2           3             1               3
AS(n):    0       0         2           3             9               9
.
Illustration of the symmetric representation of antisigma(n) = AS(n) = A024816(n), for n = 7..9:
.                                                y|          _ _ _ _
.                          y|          _ _ _      |  _ _ _ _|       |
.      y|        _ _ _      |  _ _ _  |     |     | |_       _ _    |
.       |  _ _ _|     |     | |_    | |_    |     |   |_    |_  |   |
.       | |_          |     |   |_  |_  |_ _|     |     |_    |_|  _|
.       |   |_       _|     |     |_  |_ _        |       |_      |
.       |     |_    |       |       |_    |       |         |_    |
.       |       |_  |       |         |_  |       |           |_  |
.       |         |_|       |           |_|       |             |_|
.       |_ _ _ _ _ _ _ _    |_ _ _ _ _ _ _ _ _    |_ _ _ _ _ _ _ _ _ _
.                      x                     x                       x
.
n:              7                    8                      9
a(n):           1                    2                      1
AS(n):         20                   21                     32
.
For n = 9 the figures 1, 2 and 3 below show respectively the three stages described in the Comments section as follows:
.
.   y|_ _ _ _ _ 5            y|_ _ _ _ _ _ _ _ _      y|          _ _ _ _
.    |_ _ _ _ _|              |_ _ _ _ _|       |      |  _ _ _ _|       |
.    |         |_ _ 3         | |_      |_ _ R  |      | |_       _ _    |
.    |         |_  |          |   |_    |_  |   |      |   |_    |_  |   |
.    |           |_|_ _ 5     |     |_ T  |_|_ _|      |     |_    |_|  _|
.    |               | |      |       |_      | |      |       |_      |
.    |      Q        | |      |         |_    | |      |         |_    |
.    |               | |      |    W      |_  | |      |           |_  |
.    |               | |      |             |_| |      |             |_|
.    |_ _ _ _ _ _ _ _|_|_     |_ _ _ _ _ _ _ _|_|_     |_ _ _ _ _ _ _ _ _ _
.                       x                        x                        x
.         Figure 1.                Figure 2.                Figure 3.
.         Symmetric                Symmetric                Symmetric
.       representation           representation           representation
.         of sigma(9)              of sigma(9)            of antisigma(9)
.       A000203(9) = 13          A000203(9) = 13          A024816(9) = 32
.           and of                   and of
.     Q = A024916(8) = 56      R = A004125(9) = 12
.                              T = A244048(9) = 20
.                              T = A153485(8) = 20
.                              W = A000217(8) = 36
.
Note that the symmetric representation of antisigma(9) contains a hole formed by three cells because these three cells were the central part of the symmetric representation of sigma(9).

Cf. A000203, A000217, A000290, A004125, A024816, A024916, A153485, A174973, A236104, A237270, A237271, A237593, A238443, A239660, A239931, A239932, A239933, A239934, A244048, A262259.

A244578 Sum of all aliquot divisors of all positive integers <= prime(n).

Original entry on oeis.org

1, 2, 6, 13, 33, 50, 85, 107, 155, 255, 298, 433, 523, 578, 678, 873, 1076, 1185, 1421, 1581, 1705, 1968, 2159, 2494, 3002, 3250, 3365, 3615, 3788, 4072, 5152, 5449, 5990, 6141, 7057, 7280, 7917, 8507, 8847, 9553, 10164, 10531, 11646, 11963, 12408, 12679

Offset: 1

Omar E. Pol, Jun 30 2014

nonn

Cf. A000040, A000203, A001065, A153485, A244576, A244583.

a(n) = sum(i=1, prime(n), sigma(i)-i); \\ Michel Marcus, Sep 29 2014

a(n) = A153485(A000040(n)).

a(n) ~ (Pi^2/12 - 1/2) * n^2 * log(n)^2. - Amiram Eldar, Mar 22 2024

More terms from Michel Marcus, Sep 29 2014

A328366 a(n) is the surface area of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

6, 20, 40, 70, 102, 150, 194, 256, 318, 394, 462, 566, 646, 750, 858, 984, 1088, 1238, 1354, 1518, 1666, 1826, 1966, 2182, 2344, 2532, 2720, 2944, 3120, 3384, 3572, 3826, 4054, 4298, 4534, 4860, 5084, 5356, 5624, 5964, 6212, 6572, 6832, 7176, 7512, 7840, 8124, 8564, 8874, 9260, 9608, 10012

Offset: 1

Omar E. Pol, Oct 26 2019

nonn

			For n = 1 the first level of the stepped pyramid is a cube, so a(1) = 6.

Cf. A000217, A002378, A013661, A024916, A045943, A046092, A153485, A175254 (volume), A196020, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A262626, A299692, A327329.

s=0;Do[s=s+4*DivisorSigma[1,n];t=2n(n+1);Print[(s/2)+t],{n,1,8000}] (* Metin Sariyar, Nov 20 2019 *)

from math import isqrt
def A328366(n): return (n*(n+1)<<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) = 4*A000217(n) + 2*A024916(n).
a(n) = 2*(A002378(n) + A327329(n)).
a(n) = 2*(A045943(n) + A153485(n)).
a(n) = A046092(n) + A327329(n).
a(n) = 2*A299692(n).
a(n) = c * n^2 + O(n*log(n)), where c = zeta(2) + 2 = 3.644934... . - Amiram Eldar, Mar 21 2024

A346877 Sum of the divisors, except for the largest, of the n-th odd number.

Original entry on oeis.org

0, 1, 1, 1, 4, 1, 1, 9, 1, 1, 11, 1, 6, 13, 1, 1, 15, 13, 1, 17, 1, 1, 33, 1, 8, 21, 1, 17, 23, 1, 1, 41, 19, 1, 27, 1, 1, 49, 19, 1, 40, 1, 23, 33, 1, 21, 35, 25, 1, 57, 1, 1, 87, 1, 1, 41, 1, 29, 65, 25, 12, 45, 31, 1, 47, 1, 27, 105, 1, 1, 51, 25, 35, 81, 1, 1, 81, 37

Offset: 1

Omar E. Pol, Aug 20 2021

Sum of aliquot divisors (or aliquot parts) of the n-th odd number.

a(n) has a symmetric representation.

			For n = 5 the 5th odd number is 9 and the divisors of 9 are [1, 3, 9] and the sum of the divisors of 9 except for the largest is 1 + 3 = 4, so a(5) = 4.

Bisection of A001065.
Partial sums give A347153.
Cf. A000203, A005408, A008438, A057427, A153485, A237593, A245092, A244049, A326123, A346869, A346878, A346879, A347154.

a[n_] := DivisorSigma[1, 2*n - 1] - 2*n + 1; Array[a, 100] (* Amiram Eldar, Aug 20 2021 *)
Total[Most[Divisors[#]]]&/@Range[1,161,2] (* Harvey P. Dale, Sep 29 2024 *)

a(n) = sigma(2*n-1) - (2*n-1); \\ Michel Marcus, Aug 20 2021

from sympy import divisors
def a(n): return sum(divisors(2*n-1)[:-1])
print([a(n) for n in range(1, 79)]) # Michael S. Branicky, Aug 20 2021

a(n) = A001065(2*n-1).
a(n) = A057427(n-1) + A346879(n).
G.f.: Sum_{k>=0} (2*k + 1) * x^(3*k + 2) / (1 - x^(2*k + 1)). - Ilya Gutkovskiy, Aug 20 2021
Sum_{k=1..n} a(k) = (Pi^2/8 - 1)*n^2 + O(n*log(n)). - Amiram Eldar, Mar 17 2024

cp's OEIS Frontend

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