A234515 - OEIS

%I #18 Nov 04 2024 04:24:33

%S 2,4,6,12,8,24,18,3,36,30,10,60,20,48,16,72,120,84,42,40,180,90,96,28,

%T 144,240,168,14,108,360,54,32,420,80,252,132,216,56,210,126,300,66,

%U 336,480,192,288,720,840,156,504,150,540,264,140,600,78,270,1260,432

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

%C Number k(n) = log_n(sigma(n)) = log(sigma(n)) / log(n) is number such that n^k(n) = sigma(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 A234517 - sequence of numbers a(n) such that a(n) > a(k) for all k < n.

%H Jaroslav Krizek, <a href="/A234515/b234515.txt">Table of n, a(n) for n = 1..200</a>

%e For number 2; k(2) = log_2(sigma(2)) = log_2(3) = 1,5849625007… = A020857 (maximal value of function k(n)).

%o (PARI) lista(nn=100000) = {v = vector(nn, n, if (n==1, 0, log(sigma(n))/log(n))); v = vecsort(v,,5); for (i=1, 80, print1(v[i], ", "));} \\ _Michel Marcus_, Dec 11 2014

%Y Cf. A234516, A234517, A234518, A234519, A234520, A234521, A234522, A234523, A234524.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Jan 03 2014