A214409 a(n) is the smallest conjectured m such that the irreducible fraction m/n is a known abundancy index.

1, 3, 4, 7, 6, 13, 8, 15, 13, 21, 12, 31, 14, 31, 26, 31, 18, 49, 20, 51, 32, 45, 24, 65, 31, 49, 40, 57, 30, 91, 32, 63, 52, 63, 48, 91, 38, 75, 56, 93, 42, 127, 44, 93, 88, 93, 48, 127, 57, 93, 80, 105, 54, 121, 72, 127, 80, 105, 60, 217, 62, 127, 104, 127

Offset: 1

Author

Michel Marcus, Jul 16 2012

Keywords

• nonn

Comments

The abundancy index of a number k is sigma(k)/k. When n is prime, (n+1)/n is irreducible and abund(n) = (n+1)/n, so a(n) = n + 1.

A known abundancy index is related to a limit. Terms of the sequence have been built with a limit of 10^30. So when n is composite, the values for a(n) are conjectural. A higher limit could provide smaller values.

If m < k < sigma(m) and k is relatively prime to m, then k/m is an abundancy outlaw. Hence if r/s is an abundancy index with gcd(r, s) = 1, then r >= sigma(s). [Stanton and Holdener page 3]. Since a(n) is coprime to n, this implies that a(n) >= sigma(n).

When a(n)=A214413(n), this means that a(n) is sure to be the least m satisfying the property.

Examples

			For n = 5, a(5) = 6 because 6/5 is irreducible, 6/5 is a known abundancy (namely of 5), and no number below 6 can be found with the same properties.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..1000
• Michel Marcus, Computations for k such that sigma(k)/k = a(n)/n
• Kate Moore A Geometric Representation of the Abundancy Index
• W. Nissen Augmentation of Table of Abundancies
• William G. Stanton and Judy A. Holdener, Abundancy "Outlaws" of the Form (sigma(N) + t)/N, Journal of Integer Sequences , Vol 10 (2007) , Article 07.9.6.

Crossrefs

Equal to or greater than A214413.