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Integers sorted by the sum of their divisors.
16

%I #42 Sep 10 2024 16:19:26

%S 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,

%T 27,20,26,41,43,33,35,47,34,53,28,39,49,24,38,59,61,32,67,30,46,51,55,

%U 71,73,45,57,79,44,65,83,40,58,89,36,50,42,62,69,77,52,97,101,63,103,85

%N Integers sorted by the sum of their divisors.

%C 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.

%C 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

%C By definition this is a permutation of the positive integers. Also positive integers of A299762. - _Omar E. Pol_, Mar 14 2018

%H Hugo Pfoertner, <a href="/A085790/b085790.txt">Table of n, a(n) for n=1..10000</a>

%H Max Alekseyev, <a href="https://home.gwu.edu/~maxal/gpscripts/invphi.gp">PARI/GP Scripts for Miscellaneous Math Problems: invphi.gp</a>, Oct. 2005

%H Jeppe Stig Nielsen, <a href="/A085790/a085790.txt">First 10000 rows of the triangle for a(n)</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%e 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.

%e Irregular triangle starts: (row numbers to the left are not part of the sequence)

%e n : row(n)

%e 1 : 1,

%e 2 :

%e 3 : 2,

%e 4 : 3,

%e 5 :

%e 6 : 5,

%e 7 : 4,

%e 8 : 7,

%e 9 :

%e 10 :

%e 11 :

%e 12 : 6, 11,

%e 13 : 9,

%e 14 : 13,

%e 15 : 8,

%e 16 :

%e 17 :

%e 18 : 10, 17,

%e 19 :

%e 20 : 19,

%e 21 :

%e 22 :

%e 23 :

%e 24 : 14, 15, 23,

%e 25 :

%e - _Jeppe Stig Nielsen_, Feb 02 2015, edited by _M. F. Hasler_, Nov 21 2019

%t SortBy[Table[{n,DivisorSigma[1,n]},{n,120}],Last][[;;,1]] (* _Harvey P. Dale_, Sep 10 2024 *)

%o (PARI) A085790_row(n)=invsigma(n) \\ Cf. Alekseyev link for invsigma(). - _M. F. Hasler_, Nov 21 2019

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

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

%K easy,nonn,tabf

%O 1,2

%A _Hugo Pfoertner_, Jul 23 2003