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

1,1

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

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.

Example

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.

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 *)

Crossrefs

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

Sequence in context: A010444 A010398 A010435 * A139442 A037988 A213508

Adjacent sequences: A057706 A057707 A057708 * A057710 A057711 A057712

Keyword

nonn

Author

Jack Brennen, Oct 24 2000

Extensions

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

Status

approved