A057709 Numbers k such that there is a unique m for which the sum of the aliquot parts of m (A001065) is k.

3, 4, 7, 9, 10, 11, 12, 18, 24, 26, 28, 30, 34, 36, 38, 39, 48, 56, 58, 60, 66, 68, 70, 72, 78, 80, 82, 84, 86, 94, 98, 102, 112, 116, 118, 122, 126, 128, 132, 138, 142, 144, 158, 160, 164, 168, 172, 174, 178, 180, 190, 192, 204, 208, 212, 220, 222, 224, 228, 250

Offset: 1

Author

Jack Brennen, Oct 24 2000

Keywords

• nonn

Comments

Alanen (1972) used the term "hermit" for a number k such that x = k is the only solution to A001065(x) = k. These numbers are the perfect numbers (A000396) in this sequence. Of the first 4 perfect numbers, 6, 28, 496 and 8128, only 28 is a term. - Amiram Eldar, Mar 03 2021

Examples

			12 is a member of the sequence because s(121)=12 (and because no other integer m satisfies s(m) = 12).
18 is included because the sum of aliquot parts of 289 = 1+17 = 18, this being the only number with this property. 6 is not included because the sum of aliquot parts of 6 = 1+2+3 = 6 and the sum of aliquot parts of 25 = 1+5 = 6.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Jack David Alanen, Empirical study of aliquot series, Ph.D Thesis, Yale University, 1972.
• Eric Weisstein's World of Mathematics, Restricted Divisor Function.
• Eric Weisstein's World of Mathematics, Aliquot Sequence.

Crossrefs

Cf. A000396, A001065, A005114, A048138, A057710.

Programs

Mathematica

seq[max_] := Module[{s = Table[0, {n, 1, max}], i}, Do[If[(i = DivisorSigma[1, n] - n) <= max, s[[i]]++], {n, 2, (max - 1)^2 }]; Position[s, 1] // Flatten]; seq[250] (* Amiram Eldar, Dec 26 2020 *)

Extensions

Removed 1 from the sequence. - T. D. Noe, Dec 02 2008