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 1-4 of 4 results.

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

Views

Author

Omar E. Pol, Aug 18 2021

Keywords

Comments

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

Links

Crossrefs

Partial sums of A346880.
Cf. A000203, A005843, A048050, A062731, A237593, A245092, A244049, A326124, A346869.

Programs

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

Formula

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

Views

Author

Omar E. Pol, Aug 18 2021

Keywords

Comments

a(n) has a symmetric representation.

Examples

			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.

Crossrefs

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

Programs

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

Formula

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

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

Views

Author

Omar E. Pol, Aug 20 2021

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

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

Formula

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

A347153 Sum of all divisors, except the largest of every number, of the first n odd numbers.

Original entry on oeis.org

0, 1, 2, 3, 7, 8, 9, 18, 19, 20, 31, 32, 38, 51, 52, 53, 68, 81, 82, 99, 100, 101, 134, 135, 143, 164, 165, 182, 205, 206, 207, 248, 267, 268, 295, 296, 297, 346, 365, 366, 406, 407, 430, 463, 464, 485, 520, 545, 546, 603, 604, 605, 692, 693, 694, 735, 736, 765, 830, 855
Offset: 1

Views

Author

Omar E. Pol, Aug 20 2021

Keywords

Comments

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

Crossrefs

Partial sums of A346877.
Cf. A000203, A001065, A001477, A005408, A008438, A048050, A153485, A237593, A245092, A244049, A326123, A346869, A346878, A346879, A347154.

Programs

  • Mathematica
    s[n_] := DivisorSigma[1, 2*n - 1] - 2*n + 1; Accumulate @ Array[s, 100] (* Amiram Eldar, Aug 20 2021 *)
  • PARI
    a(n) = sum(k=1, n, k = 2*k-1; sigma(k)-k); \\ Michel Marcus, Aug 20 2021
  • Python
    from sympy import divisors
from itertools import accumulate
def A346877(n): return sum(divisors(2*n-1)[:-1])
def aupton(nn): return list(accumulate(A346877(n) for n in range(1, nn+1)))
print(aupton(60)) # Michael S. Branicky, Aug 20 2021

Formula

a(n) = A001477(n-1) + A346869(n).
G.f.: (1/(1 - x)) * Sum_{k>=0} (2*k + 1) * x^(3*k + 2) / (1 - x^(2*k + 1)). - Ilya Gutkovskiy, Aug 20 2021
a(n) = (Pi^2/8 - 1)*n^2 + O(n*log(n)). - Amiram Eldar, Mar 21 2024
Showing 1-4 of 4 results.

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