cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A062731 Sum of divisors of 2*n.

Original entry on oeis.org

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

Views

Author

Jason Earls, Jul 11 2001

Keywords

Comments

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

Links

Crossrefs

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.

Programs

Formula

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)

Extensions

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

Views

Author

Omar E. Pol, Aug 20 2021

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    a[n_] := DivisorSigma[1, 2*n] - 2*n; Array[a, 100] (* Amiram Eldar, Aug 20 2021 *)
  • PARI
    a(n) = sigma(2*n) - 2*n; \\ Michel Marcus, Aug 20 2021
  • Python
    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

Formula

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

A346880 Sum of the divisors, except the smallest and the largest, of the n-th positive even number.

Original entry on oeis.org

0, 2, 5, 6, 7, 15, 9, 14, 20, 21, 13, 35, 15, 27, 41, 30, 19, 54, 21, 49, 53, 39, 25, 75, 42, 45, 65, 63, 31, 107, 33, 62, 77, 57, 73, 122, 39, 63, 89, 105, 43, 139, 45, 91, 143, 75, 49, 155, 72, 116, 113, 105, 55, 171, 105, 135, 125, 93, 61, 239, 63, 99, 185, 126, 121
Offset: 1

Views

Author

Omar E. Pol, Aug 18 2021

Keywords

Comments

a(n) has a symmetric representation.

Examples

			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 the smaller and the largest is 2 + 5 = 7, so a(5) = 7.

Crossrefs

Bisection of A048050.
Partial sums give A346870.
Cf. A000203, A005843, A062731, A237593, A245092, A244049, A326124, A346879.

Programs

  • Mathematica
    a[n_] := DivisorSigma[1, 2*n] - 2*n - 1; Array[a, 100] (* Amiram Eldar, Aug 19 2021 *)
  • Python
    from sympy import divisors
def a(n): return sum(divisors(2*n)[1:-1])
print([a(n) for n in range(1, 66)]) # Michael S. Branicky, Aug 19 2021

Formula

a(n) = A048050(2*n).
Sum_{k=1..n} a(k) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, Mar 21 2024

A346869 Sum of all divisors, except the smallest and the largest of every number, of the first n odd numbers.

Original entry on oeis.org

0, 0, 0, 0, 3, 3, 3, 11, 11, 11, 21, 21, 26, 38, 38, 38, 52, 64, 64, 80, 80, 80, 112, 112, 119, 139, 139, 155, 177, 177, 177, 217, 235, 235, 261, 261, 261, 309, 327, 327, 366, 366, 388, 420, 420, 440, 474, 498, 498, 554, 554, 554, 640, 640, 640, 680, 680, 708, 772, 796
Offset: 1

Views

Author

Omar E. Pol, Aug 18 2021

Keywords

Comments

Partial sums of the odd-indexed terms of Chowla's function A048050.
a(n) has a symmetric representation.

Crossrefs

Partial sums of A346879.
Cf. A000203, A005408, A008438, A048050, A237593, A245092, A244049, A326123, A346870.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=1, 0,
      a(n-1)+numtheory[sigma](2*n-1)-2*n)
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 19 2021
  • Mathematica
    s[1] = 0; s[n_] := DivisorSigma[1, 2*n - 1] - 2*n; Accumulate @ Array[s, 50] (* Amiram Eldar, Aug 19 2021 *)
Accumulate[Join[{0},Table[DivisorSigma[1,n]-n-1,{n,3,151,2}]]] (* Harvey P. Dale, Jul 29 2023 *)
  • Python
    from sympy import divisors
from itertools import accumulate
def A346879(n): return sum(divisors(2*n-1)[1:-1])
def aupton(nn): return list(accumulate(A346879(n) for n in range(1, nn+1)))
print(aupton(60)) # Michael S. Branicky, Aug 19 2021

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

Views

Author

Omar E. Pol, Aug 20 2021

Keywords

Comments

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.

Links

Crossrefs

Partial sums of A346878.
Cf. A000203, A005843, A048050, A062731, A237593, A245092, A244049, A326124, A346870, A346877, A346880, A347153.

Programs

  • Mathematica
    s[n_] := DivisorSigma[1, 2*n] - 2*n; Accumulate @ Array[s, 100] (* Amiram Eldar, Aug 20 2021 *)
  • PARI
    a(n) = sum(k=1, n, k*=2; sigma(k)-k); \\ Michel Marcus, Aug 20 2021
  • Python
    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
  • Python
    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

Formula

a(n) = n + A346870(n).
a(n) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, May 15 2023
Showing 1-5 of 5 results.

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