A348711 - OEIS

%I #20 Nov 11 2021 10:43:40

%S 1,2,3,4,6,5,8,12,10,7,9,14,16,18,15,20,24,11,13,30,22,21,28,36,17,26,

%T 32,40,19,27,42,48,25,34,60,23,33,44,38,54,35,45,56,72,50,39,52,29,46,

%U 31,66,84,64,70,80,90,37,51,78,96,68,58,63,120,41,62,57,76,108,55

%N Integers sorted by decreasing value of sigma(x)/x^2, where sigma is the sum of divisors.

%C Is it possible to find distinct x and y such that sigma(x)/x^2 = sigma(y)/y^2 ?

%H Michel Marcus, <a href="/A348711/b348711.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A348711/a348711.png">Scatterplot of sigma(x)/x^2</a> for x=1..240, highlighting the position of superabundant x (in A004394) in red.

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

%e For 1,2,3,4,6,5,... sigma(x)/x^2 gives: 1 > 3/4 > 4/9 > 7/16 > 1/3 > 6/25 > ...

%t Block[{nn = Function[s, FirstPosition[s, #][[1]] &@ Fold[Max, s]]@ Array[DivisorSigma[1, #]/# &, 125], s}, s = Array[DivisorSigma[1, #]/#^2 &, nn]; #[[1 ;; FirstPosition[#, nn][[1]]]] &@ Map[FirstPosition[s, #][[1]] &, ReverseSort[s]]] (* _Michael De Vlieger_, Oct 31 2021 *)

%o (PARI) lista(nn) = Vec(vecsort(vector(10*nn, k, -sigma(k)/k^2), , 1), nn);

%Y Cf. A000203, A004394, A085790.

%K nonn

%O 1,2

%A _Michel Marcus_, Oct 31 2021