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, 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

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: A340069 A265352 A265368 * A162425 A217254 A223488

Adjacent sequences: A239969 A239970 A239971 * A239973 A239974 A239975

Keyword

nonn

Author

Michel Marcus, Mar 30 2014

Status

approved