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%I #7 Apr 30 2022 10:35:40
%S 4,4,13,3,32,45,63,90,111,162,210,237,7502,11776,240,2944,3008,11648,
%T 225152,478080,2704000,1548160,2462400,10128256,4999296,8640768,
%U 48264064,32347648
%N a(n) is the numerator of the median abundancy index of the first 2^n + 1 positive integers.
%C The abundancy index of a number k is sigma(k)/k = A017665(k)/A017666(k), where sigma is the sum-of-divisors function (A000203).
%F Limit_{n->oo} a(n)/A353616(n) = A353617.
%e The fractions begin with 4/3, 4/3, 13/9, 3/2, 32/21, 45/29, 63/41, 90/59, 111/73, 162/107, ...
%e The abundancy indices of the first 3 positive integers are 1, 3/2 and 4/3. The median is a(1)/A353616(1) = 4/3.
%t With[{m = 20}, t = Table[DivisorSigma[-1, n], {n, 1, 2^m + 1}]; Numerator @ Table[Median[t[[1 ;; 2^n + 1]]], {n, 1, m}]]
%Y Cf. A000203, A017665, A017666, A353616 (denominators), A353617.
%K nonn,frac,more
%O 1,1
%A _Amiram Eldar_, Apr 30 2022