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 1-7 of 7 results.

A385148 a(n) = A001065(A346878(n)).

Original entry on oeis.org

0, 1, 6, 1, 7, 15, 8, 9, 11, 14, 10, 55, 15, 28, 54, 1, 22, 17, 14, 43, 66, 50, 16, 64, 1, 26, 78, 63, 31, 172, 20, 41, 90, 32, 40, 45, 50, 63, 144, 56, 40, 196, 26, 76, 259, 64, 43, 236, 1, 65, 126, 56, 64, 136, 56, 134, 186, 50, 34, 504, 63, 117, 198, 1, 64, 300, 74, 70, 222, 203
Offset: 1

Views

Author

Michel Marcus, Jun 19 2025

Keywords

Comments

There are only 2 known fixed points 26 and 296; they are the numbers k such that k = sigma(m) - m where m = sigma(2*k) - 2*k as investigated by S. I. Dimitrov. See link.

Links

Crossrefs

Cf. A001065, A346878.
Cf. A377766 (twice the integers that satisfy a(n)=1).
Cf. A384411.

Programs

  • Mathematica
    f[x_] := DivisorSigma[1, x] - x; Table[Nest[f, 2*n, 2], {n, 120}] (* Michael De Vlieger, Jun 19 2025 *)
  • PARI
    a(n) = my(m=sigma(2*n) - 2*n); sigma(m) - m ;

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

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

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

A359079 a(n) is the sum of the divisors d of 2*n such that the binary expansions of d and 2*n have no common 1-bit.

Original entry on oeis.org

1, 3, 1, 7, 6, 6, 1, 15, 10, 13, 1, 16, 1, 3, 1, 31, 18, 33, 1, 32, 22, 3, 1, 36, 6, 3, 10, 14, 1, 6, 1, 63, 34, 54, 1, 70, 38, 22, 1, 70, 42, 48, 1, 7, 6, 3, 1, 76, 1, 38, 18, 7, 1, 24, 1, 36, 1, 3, 1, 21, 1, 3, 1, 127, 84, 116, 1, 126, 70, 38, 1, 153, 74, 77
Offset: 1

Views

Author

Rémy Sigrist, Dec 15 2022

Keywords

Comments

Odd numbers share a 1-bit (2^0) with all their divisors, hence this sequence deals with even numbers.

Examples

			For n = 6:
- the divisors of 12 are:
      d   bin(d)  common bit?
      --  ------  -----------
       1       1  no
       2      10  no
       3      11  no
       4     100  yes
       6     110  yes
      12    1100  yes
- hence a(6) = 1 + 2 + 3 = 6.

Links

Crossrefs

Cf. A246601, A346878.

Programs

  • Mathematica
    a[n_] := DivisorSum[2n, #*Boole[BitAnd[#, 2n] == 0] &]; Array[a, 74]
  • PARI
    a(n) = sumdiv(2*n, d, if (bitand(2*n,d)==0, d, 0))
  • Python
    from sympy import divisors as divs
def a(n): return sum(d for d in divs(2*n, generator=True) if (d>>1)&n == 0)
print([a(n) for n in range(1, 75)]) # Michael S. Branicky, Dec 15 2022

Formula

a(n) <= A346878(n) with equality iff n is a power of 2.

A384411 Pairs (k, m) such that k = sigma(m) - m and m = sigma(2*k) - 2*k.

Original entry on oeis.org

26, 46, 296, 586
Offset: 1

Views

Author

S. I. Dimitrov, Jun 01 2025

Keywords

Comments

Fixed points of x->A346878(A001065(x)).
Next term > 15*10^7.

Examples

			(26, 46) is such a pair because 26 = sigma(46) - 46 and 46 = sigma(52) - 52.
(296, 586) is another pair.

Links

Crossrefs

Cf. A000203, A001065, A063990, A259180, A346878, A383483.

Programs

  • PARI
    for(k=1,10^9, m = sigma(2*k) - 2*k;if(k == sigma(m) - m, print1(k, ", ", m, ", "))); \\ Joerg Arndt, Jun 01 2025
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.