A346880 -id:A346880 - cp's OEIS frontend

A062731 Sum of divisors of 2*n.

3, 7, 12, 15, 18, 28, 24, 31, 39, 42, 36, 60, 42, 56, 72, 63, 54, 91, 60, 90, 96, 84, 72, 124, 93, 98, 120, 120, 90, 168, 96, 127, 144, 126, 144, 195, 114, 140, 168, 186, 126, 224, 132, 180, 234, 168, 144, 252, 171, 217, 216, 210, 162, 280, 216, 248, 240, 210

Offset: 1

Jason Earls, Jul 11 2001

a(n) is also the total number of parts in all partitions of 2*n into equal parts. - Omar E. Pol, Feb 14 2021

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 1000 terms from Harry J. Smith]

Sigma(k*n): A000203 (k=1), A144613 (k=3), A193553 (k=4, even bisection), A283118 (k=5), A224613 (k=6), A283078 (k=7), A283122 (k=8), A283123 (k=9).
Cf. A008438, A074400, A182818, A239052 (odd bisection), A326124 (partial sums), A054784, A215947, A336923, A346870, A346878, A346880, A355750.
Row 2 of A319526. Column & Row 2 of A216626. Row 1 of A355927.
Shallow diagonal (2n,n) of A265652. See also A244658.

[SumOfDivisors(2*n): n in [1..70]]; // Vincenzo Librandi, Oct 31 2014

lst={};Do[AppendTo[lst, DivisorSigma[1, n]], {n, 2, 6!, 2}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 20 2008 *)
DivisorSigma[1,2*Range[60]] (* Harvey P. Dale, Jun 08 2022 *)

numlib::sigma(2*n)$ n=0..81 // Zerinvary Lajos, May 13 2008

PARI
```
vector(66,n,sigma(2*n,1))
```

for (n=1, 1000, write("b062731.txt", n, " ", sigma(2*n)) ) \\ Harry J. Smith, Aug 09 2009

a(n) = A000203(2*n). - R. J. Mathar, Apr 06 2011
a(n) = A000203(n) + A054785(n). - R. J. Mathar, May 19 2020
From Vaclav Kotesovec, Aug 07 2022: (Start)
Dirichlet g.f.: zeta(s) * zeta(s-1) * (3 - 2^(1-s)).
Sum_{k=1..n} a(k) ~ 5 * Pi^2 * n^2 / 24. (End)
From Miles Wilson, Sep 30 2024: (Start)
G.f.: Sum_{k>=1} k*x^(k/gcd(k, 2))/(1 - x^(k/gcd(k, 2))).
G.f.: Sum_{k>=1} k*x^(2*k/(3 + (-1)^k))/(1 - x^(2*k/(3 + (-1)^k))). (End)

Zero removed and offset corrected by Omar E. Pol, Jul 17 2009

A346878 Sum of the divisors, except for the largest, of the n-th positive even number.

Original entry on oeis.org

1, 3, 6, 7, 8, 16, 10, 15, 21, 22, 14, 36, 16, 28, 42, 31, 20, 55, 22, 50, 54, 40, 26, 76, 43, 46, 66, 64, 32, 108, 34, 63, 78, 58, 74, 123, 40, 64, 90, 106, 44, 140, 46, 92, 144, 76, 50, 156, 73, 117, 114, 106, 56, 172, 106, 136, 126, 94, 62, 240, 64, 100, 186, 127

Offset: 1

Omar E. Pol, Aug 20 2021

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

a(n) has a symmetric representation.

			For n = 5 the 5th even number is 10 and the divisors of 10 are [1, 2, 5, 10] and the sum of the divisors of 10 except for the largest is 1 + 2 + 5 = 8, so a(5) = 8.

Bisection of A001065.
Partial sums give A347154.
Cf. A000203, A005843, A048050, A062731, A237593, A245092, A244049, A326124, A346870, A346877, A346880, A347153.

a[n_] := DivisorSigma[1, 2*n] - 2*n; Array[a, 100] (* Amiram Eldar, Aug 20 2021 *)

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

from sympy import divisors
def a(n): return sum(divisors(2*n)[:-1])
print([a(n) for n in range(1, 65)]) # Michael S. Branicky, Aug 20 2021

a(n) = A001065(2*n).
a(n) = 1 + A346880(n).
Sum_{k=1..n} a(k) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, Mar 17 2024

A346870 Sum of all divisors, except the smallest and the largest of every number, of the first n positive even numbers.

Original entry on oeis.org

0, 2, 7, 13, 20, 35, 44, 58, 78, 99, 112, 147, 162, 189, 230, 260, 279, 333, 354, 403, 456, 495, 520, 595, 637, 682, 747, 810, 841, 948, 981, 1043, 1120, 1177, 1250, 1372, 1411, 1474, 1563, 1668, 1711, 1850, 1895, 1986, 2129, 2204, 2253, 2408, 2480, 2596, 2709, 2814

Offset: 1

Omar E. Pol, Aug 18 2021

Partial sums of the even-indexed terms of Chowla's function A048050.

a(n) has a symmetric representation.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Partial sums of A346880.

Cf. A000203, A005843, A048050, A062731, A237593, A245092, A244049, A326124, A346869.

a:= proc(n) option remember; `if`(n=0, 0,
      a(n-1)+numtheory[sigma](2*n)-1-2*n)
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 19 2021

s[n_] := DivisorSigma[1, 2*n] - 2*n - 1; Accumulate @ Array[s, 50] (* Amiram Eldar, Aug 19 2021 *)

from sympy import divisors
from itertools import accumulate
def A346880(n): return sum(divisors(2*n)[1:-1])
def aupton(nn): return list(accumulate(A346880(n) for n in range(1, nn+1)))
print(aupton(52)) # Michael S. Branicky, Aug 19 2021

from math import isqrt
def A346870(n): return (t:=isqrt(m:=n>>1))**2*(t+1) - sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))-3*((s:=isqrt(n))**2*(s+1) - sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1)-n*(n+2) # Chai Wah Wu, Nov 02 2023

a(n) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, May 15 2023

A346879 Sum of the divisors, except the smallest and the largest, of the n-th odd number.

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 0, 8, 0, 0, 10, 0, 5, 12, 0, 0, 14, 12, 0, 16, 0, 0, 32, 0, 7, 20, 0, 16, 22, 0, 0, 40, 18, 0, 26, 0, 0, 48, 18, 0, 39, 0, 22, 32, 0, 20, 34, 24, 0, 56, 0, 0, 86, 0, 0, 40, 0, 28, 64, 24, 11, 44, 30, 0, 46, 0, 26, 104, 0, 0, 50, 24, 34, 80, 0, 0, 80, 36

Offset: 1

Omar E. Pol, Aug 18 2021

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 the smaller and the largest is 3, so a(5) = 3.
For n = 6 the 6th odd number is 11 and the divisors of 11 are [1, 11] and the sum of the divisors of 11 except the smaller and the largest is 0, so a(6) = 0.

Bisection of A048050.
Partial sums give A346869.
Cf. A000203, A005408, A008438, A237593, A245092, A244049, A326123, A346880.

a[1] = 0; a[n_] := DivisorSigma[1, 2*n - 1] - 2*n; Array[a, 100] (* Amiram Eldar, Aug 19 2021 *)

from sympy import divisors
def a(n): return sum(divisors(2*n-1)[1:-1])
print([a(n) for n in range(1, 79)]) # Michael S. Branicky, Aug 19 2021

a(n) = A048050(2*n-1).

A347154 Sum of all divisors, except the largest of every number, of the first n positive even numbers.

Original entry on oeis.org

1, 4, 10, 17, 25, 41, 51, 66, 87, 109, 123, 159, 175, 203, 245, 276, 296, 351, 373, 423, 477, 517, 543, 619, 662, 708, 774, 838, 870, 978, 1012, 1075, 1153, 1211, 1285, 1408, 1448, 1512, 1602, 1708, 1752, 1892, 1938, 2030, 2174, 2250, 2300, 2456, 2529, 2646, 2760

Offset: 1

Omar E. Pol, Aug 20 2021

Sum of all aliquot divisors (or aliquot parts) of the first n positive even numbers.
Partial sums of the even-indexed terms of A001065.
a(n) has a symmetric representation.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Partial sums of A346878.

Cf. A000203, A005843, A048050, A062731, A237593, A245092, A244049, A326124, A346870, A346877, A346880, A347153.

s[n_] := DivisorSigma[1, 2*n] - 2*n; Accumulate @ Array[s, 100] (* Amiram Eldar, Aug 20 2021 *)

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

from sympy import divisors
from itertools import accumulate
def A346878(n): return sum(divisors(2*n)[:-1])
def aupton(nn): return list(accumulate(A346878(n) for n in range(1, nn+1)))
print(aupton(51)) # Michael S. Branicky, Aug 20 2021

from math import isqrt
def A347154(n): return (t:=isqrt(m:=n>>1))**2*(t+1) - sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))-3*((s:=isqrt(n))**2*(s+1) - sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1)-n*(n+1) # Chai Wah Wu, Nov 02 2023

a(n) = n + A346870(n).

a(n) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, May 15 2023

A348554 Irregular triangle read by rows: row n gives the divisors d of 2n with 1 < d < 2n, for n >= 2.

Original entry on oeis.org

2, 2, 3, 2, 4, 2, 5, 2, 3, 4, 6, 2, 7, 2, 4, 8, 2, 3, 6, 9, 2, 4, 5, 10, 2, 11, 2, 3, 4, 6, 8, 12, 2, 13, 2, 4, 7, 14, 2, 3, 5, 6, 10, 15, 2, 4, 8, 16, 2, 17, 2, 3, 4, 6, 9, 12, 18, 2, 19, 2, 4, 5, 8, 10, 20, 2, 3, 6, 7, 14, 21, 2, 4, 11, 22, 2, 23, 2, 3, 4, 6, 8, 12, 16, 24, 2, 5, 10, 25

Offset: 2

Wolfdieter Lang, Oct 22 2021

This gives the rows 2*n of A137510, for n >= 2.
The length of row n is A069930(n) = tau(2*n) - 2 = A099777(n) - 2.
The sum of row n is A346880(n) = A062731(n) - (2*n + 1).

			The irregular triangle T(n, k) begins:
n, 2*n / k 1  2  3  4  5  6  7 ...
----------------------------------
2,   4:    2
3,   6:    2  3
4,   8:    2  4
5,  10:    2  5
6   12:    2  3  4 6
7,  14:    2  7
8,  16:    2  4  8
9,  18:    2  3  6  9
10, 20:    2  4  5 10
11, 22:    2 11
12, 24:    2  3  4  6  8 12
13, 26:    2 13
14, 28:    2  4  7 14
15, 30:    2  3  5  6 10 15
16, 32:    2  4  8  1
17, 34:    2 17
18, 36:    2  3  4  6  9 12 18
19, 38:    2 19
20, 40:    2  4  5  8 10 20
...

Cf. A069930, A099777, A137510, A346880.

Flatten@Table[Select[Divisors[2n],1<#<2n&],{n,2,25}] (* Giorgos Kalogeropoulos, Oct 22 2021 *)

row(n) = select(x->((x>1) && (x<2*n)), divisors(2*n)); \\ Michel Marcus, Oct 23 2021

T(n, k) = A137510(2*n, k), for n >= 2 and k = 1, 2, ..., A069930(n).

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A062731 Sum of divisors of 2*n.

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A346878 Sum of the divisors, except for the largest, of the n-th positive even number.

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A346870 Sum of all divisors, except the smallest and the largest of every number, of the first n positive even numbers.

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A346879 Sum of the divisors, except the smallest and the largest, of the n-th odd number.

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A347154 Sum of all divisors, except the largest of every number, of the first n positive even numbers.

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Author

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Mathematica

PARI

Python

Python

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A348554 Irregular triangle read by rows: row n gives the divisors d of 2*n with 1 < d < 2*n, for n >= 2.

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Mathematica

PARI

Formula

A348554 Irregular triangle read by rows: row n gives the divisors d of 2n with 1 < d < 2n, for n >= 2.