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

Author

Hugo Pfoertner, Jul 23 2003

Keywords

• easy
• nonn
• tabf

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

Examples

			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
		

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

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)).

Programs

Mathematica

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

PARI

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