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Natural numbers n sorted by decreasing values of number k(n) = sigma(n)^(1/n), where sigma(n) = A000203(n) = the sum of divisors of n.
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%I #6 Jan 14 2014 12:45:11

%S 2,4,3,6,5,8,7,10,9,12,14,11,16,15,18,13,20,24,17,21,22,19,28,26,30,

%T 23,25,27,32,36,34,33,29,40,31,35,42,38,39,44,48,37,45,46,41,50,54,52,

%U 43,56,60,51,49,47,55,58,57,64,66,53,63,62,72,68,70,59,65

%N Natural numbers n sorted by decreasing values of number k(n) = sigma(n)^(1/n), where sigma(n) = A000203(n) = the sum of divisors of n.

%C Number k(n) = sigma(n)^(1/n) is number such that k(n)^n = sigma(n).

%C For number 2; k(2) = sigma(2)^(1/2) = sqrt(3) = 1,732050807568… = A002194 (maximal value of function k(n)).

%C The last term of this infinite sequence is number 1, k(1) = 1 (minimal value of function k(n)).

%C Conjecture: Every natural number n has a unique value of number k(n).

%C See A234521 - sequence of numbers a(n) such that a(n) > a(k) for all k < n.

%H Jaroslav Krizek, <a href="/A234519/b234519.txt">Table of n, a(n) for n = 1..1000</a>

%o (PARI) a(n)=vecsort(vector(2*n, i, sigma(i)^(1/i)), ,5)[n] \\ _Michel Marcus_ and _Ralf Stephan_, Jan 14 2014

%Y Cf. A234515, A234516, A234517, A234518, A234520, A234521, A234522, A234523, A234524.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Jan 04 2014