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A085790 Integers sorted by the sum of their divisors.

%I A085790 #42 Sep 10 2024 16:19:26
%S A085790 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 A085790 27,20,26,41,43,33,35,47,34,53,28,39,49,24,38,59,61,32,67,30,46,51,55,
%U A085790 71,73,45,57,79,44,65,83,40,58,89,36,50,42,62,69,77,52,97,101,63,103,85
%N A085790 Integers sorted by the sum of their divisors.
%C A085790 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 A085790 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 A085790 By definition this is a permutation of the positive integers. Also positive integers of A299762. - _Omar E. Pol_, Mar 14 2018
%H A085790 Hugo Pfoertner, <a href="/A085790/b085790.txt">Table of n, a(n) for n=1..10000</a>
%H A085790 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 A085790 Jeppe Stig Nielsen, <a href="/A085790/a085790.txt">First 10000 rows of the triangle for a(n)</a>
%H A085790 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A085790 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 A085790 Irregular triangle starts: (row numbers to the left are not part of the sequence)
%e A085790    n : row(n)
%e A085790    1 : 1,
%e A085790    2 :
%e A085790    3 : 2,
%e A085790    4 : 3,
%e A085790    5 :
%e A085790    6 : 5,
%e A085790    7 : 4,
%e A085790    8 : 7,
%e A085790    9 :
%e A085790   10 :
%e A085790   11 :
%e A085790   12 : 6, 11,
%e A085790   13 : 9,
%e A085790   14 : 13,
%e A085790   15 : 8,
%e A085790   16 :
%e A085790   17 :
%e A085790   18 : 10, 17,
%e A085790   19 :
%e A085790   20 : 19,
%e A085790   21 :
%e A085790   22 :
%e A085790   23 :
%e A085790   24 : 14, 15, 23,
%e A085790   25 :
%e A085790 - _Jeppe Stig Nielsen_, Feb 02 2015, edited by _M. F. Hasler_, Nov 21 2019
%t A085790 SortBy[Table[{n,DivisorSigma[1,n]},{n,120}],Last][[;;,1]] (* _Harvey P. Dale_, Sep 10 2024 *)
%o A085790 (PARI) A085790_row(n)=invsigma(n) \\ Cf. Alekseyev link for invsigma(). - _M. F. Hasler_, Nov 21 2019
%Y A085790 Cf. A000203 (sigma), A007609 (values taken by sigma, with multiplicity), A002191 (possible values for sigma), A002192 (first column).
%Y A085790 Cf. A152454 (similar sequence for proper divisors only (aliquot parts)).
%K A085790 easy,nonn,tabf
%O A085790 1,2
%A A085790 _Hugo Pfoertner_, Jul 23 2003