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%I #16 Dec 29 2016 03:15:19
%S 2,3,4,7,7,7,8,16,13,11,13,13,16,16,16,21,19,19,21,21,27,23,25,25,27,
%T 27,32,29,31,31,32,36,34,35,36,50,39,39,49,41,43,43,47,45,46,47,49,49,
%U 50,63,52,53,55,55,57,57,63,59,61,61,63,63,64,93,75,67,71
%N a(n) is the smallest m greater than n such that both the numerator and the denominator of the abundancy index of m are greater than those of n.
%C Recall that the abundancy index of n is defined as sigma(n)/n with sigma(n) being the sum of divisors of n (A000203). The numerators and denominators of the abundancy index can be found in A017665 and A017666.
%C This sequence is not monotonic; for instance, a(9)=13 is smaller than a(8)=16. See sequence A239973 for records.
%H Ivan Neretin, <a href="/A239972/b239972.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Abundancy.html">Abundancy</a>
%e For n=1 to 8 the abundancy indices are 1/1, 3/2, 4/3, 7/4, 6/5, 2/1, 8/7, 15/8.
%e a(1)=2 because, comparing 1/1 and 3/2, we have both 1<3 and 1<2.
%e a(4)=7 because, comparing 7/4 and 8/7, we have both 7<8 and 4<7.
%t a = abun = {}; Do[k = 1; While[True, If[k > Length@abun, AppendTo[abun, DivisorSigma[1, n + k - 1]/(n + k - 1)]]; If[Numerator[abun[[k]]] > Numerator[abun[[1]]] && Denominator[abun[[k]]] > Denominator[abun[[1]]], Break[]]; k++]; AppendTo[a, n + k - 1]; abun = Delete[abun, 1], {n, 67}]; a (* _Ivan Neretin_, Dec 28 2016 *)
%o (PARI) a(n) = {my(ab = sigma(n)/n); my(num = numerator(ab)); my(den = denominator(ab)); my(ok = 0); my(m = n); while (! ok, m++; ab = sigma(m)/m; nab = numerator(ab); dab = denominator(ab); if ((nab > num) && (dab > den), ok = 1);); m;}
%Y Cf. A000203, A017665, A017666, A239973.
%K nonn
%O 1,1
%A _Michel Marcus_, Mar 30 2014
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