A085790 Integers sorted by the sum of their divisors.

1, 2, 3, 5, 4, 7, 6, 11, 9, 13, 8, 10, 17, 19, 14, 15, 23, 12, 29, 16, 25, 21, 31, 22, 37, 18, 27, 20, 26, 41, 43, 33, 35, 47, 34, 53, 28, 39, 49, 24, 38, 59, 61, 32, 67, 30, 46, 51, 55, 71, 73, 45, 57, 79, 44, 65, 83, 40, 58, 89, 36, 50, 42, 62, 69, 77, 52, 97, 101, 63, 103, 85

Offset

1,2

Comments

Integers having the same sum of divisors are sorted in ascending order, e.g., sigma(14)=sigma(15)=sigma(23)=24 -> a(15)=14, a(16)=15, a(17)=23.

Also an irregular triangle where the k-th row consists of all numbers with divisor sum k. See A054973(k) for the k-th row length. - Jeppe Stig Nielsen, Jan 29 2015

By definition this is a permutation of the positive integers. Also positive integers of A299762. - Omar E. Pol, Mar 14 2018

Links

Hugo Pfoertner, Table of n, a(n) for n=1..10000

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invphi.gp, Oct. 2005

Jeppe Stig Nielsen, First 10000 rows of the triangle for a(n)

Index entries for sequences that are permutations of the natural numbers

Example

a(9) = 9, a(10) = 13, a(11) = 8 because sigma(9) = 9 + 3 + 1 = 13, sigma(13) = 13 + 1 = 14, sigma(8) = 8 + 4 + 2 + 1 = 15 and there are no other numbers with those sigma values.
Irregular triangle starts: (row numbers to the left are not part of the sequence)
   n : row(n)
   1 : 1,
   2 :
   3 : 2,
   4 : 3,
   5 :
   6 : 5,
   7 : 4,
   8 : 7,
   9 :
  10 :
  11 :
  12 : 6, 11,
  13 : 9,
  14 : 13,
  15 : 8,
  16 :
  17 :
  18 : 10, 17,
  19 :
  20 : 19,
  21 :
  22 :
  23 :
  24 : 14, 15, 23,
  25 :
- Jeppe Stig Nielsen, Feb 02 2015, edited by M. F. Hasler, Nov 21 2019

Mathematica

SortBy[Table[{n, DivisorSigma[1, n]}, {n, 120}], Last][[;; , 1]] (* Harvey P. Dale, Sep 10 2024 *)

Prog

(PARI) A085790_row(n)=invsigma(n) \\ Cf. Alekseyev link for invsigma(). - M. F. Hasler, Nov 21 2019

Crossrefs

Cf. A000203 (sigma), A007609 (values taken by sigma, with multiplicity), A002191 (possible values for sigma), A002192 (first column).

Cf. A152454 (similar sequence for proper divisors only (aliquot parts)).

Sequence in context: A282649 A102454 A232640 * A257339 A210882 A117120

Adjacent sequences: A085787 A085788 A085789 * A085791 A085792 A085793

Keyword

easy,nonn,tabf

Author

Hugo Pfoertner, Jul 23 2003

Status

approved