

A239972


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.


2



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, 27, 32, 29, 31, 31, 32, 36, 34, 35, 36, 50, 39, 39, 49, 41, 43, 43, 47, 45, 46, 47, 49, 49, 50, 63, 52, 53, 55, 55, 57, 57, 63, 59, 61, 61, 63, 63, 64, 93, 75, 67, 71
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OFFSET

1,1


COMMENTS

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.
This sequence is not monotonic; for instance, a(9)=13 is smaller than a(8)=16. See sequence A239973 for records.


LINKS

Ivan Neretin, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Abundancy


EXAMPLE

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.
a(1)=2 because, comparing 1/1 and 3/2, we have both 1<3 and 1<2.
a(4)=7 because, comparing 7/4 and 8/7, we have both 7<8 and 4<7.


MATHEMATICA

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 *)


PROG

(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; }


CROSSREFS

Cf. A000203, A017665, A017666, A239973.
Sequence in context: A290641 A265352 A265368 * A162425 A217254 A223488
Adjacent sequences: A239969 A239970 A239971 * A239973 A239974 A239975


KEYWORD

nonn


AUTHOR

Michel Marcus, Mar 30 2014


STATUS

approved



