A087998 a(n) = smallest number x such that sigma(x) = 2x + 2n.

6, 20, 12, 8925, 56, 40, 24, 272, 550, 208, 176, 1312, 112, 80, 48, 945, 572, 928, 2205, 5696, 736, 9555, 350, 490, 60, 416, 352, 90, 84, 160, 96, 24704, 108, 3904, 260, 487936, 132, 1575, 340, 234, 156, 22144, 2752, 2624, 460, 306, 500, 475648, 204

Offset: 0

Author

Labos Elemer, Oct 13 2003

Keywords

• nonn

Comments

The satellite problem: sigma(x)=2x+odd seems much more difficult.

Solutions (square or twice a square!) obtained only for: 3,7,17,19,31,39,41,51,59,65,71,89,115,119,127,161,185,199 (see A140863).

a(221) <= 576460514469609472. - Donovan Johnson, Jan 06 2014

Examples

			n=67: 2n=134, first solution to sigma(x)=2x+134 is a(67)=1958912;
n=0: solution is the least perfect number, a(0)=6;
2n=12, 2n=56 provide large number of solutions.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 0..220
• Nichole Davis, Dominic Klyve and Nicole Kraght, On the difference between an integer and the sum of its proper divisors, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493.

Programs

Mathematica

ds[x_, de_] := DivisorSigma[1, x]-2*x-de a[n_] := Block[{m=1, s=ds[m, n]}, While[(s !=0)&& !Greater[m, 10000000], m++ ]; m]; Table[a[n], {n, 1, 100}]//Timing

Extensions

a(47) corrected by Donovan Johnson, Jan 06 2014