A083211 Abundant numbers (A005101) with no subset of their divisors such that the complement has the same sum.

18, 36, 72, 100, 144, 162, 196, 200, 288, 324, 392, 400, 450, 576, 648, 738, 748, 774, 784, 800, 846, 882, 900, 954, 968, 1062, 1098, 1152, 1296, 1352, 1458, 1568, 1600, 1764, 1800, 1936, 2178, 2500, 2592, 2704, 2916, 3042, 3136, 3200, 3528, 3600, 3872

Offset

1,1

Comments

A083206(a(n)) = 0; subsequence of A083210.

All numbers with an odd sum of divisors (either a square or twice a square, A028982) must be terms because for these numbers the two subsets will be of opposite parity. - Robert G. Wilson v, Apr 01 2010

Links

Table of n, a(n) for n=1..47.

Eric Weisstein's World of Mathematics, Abundant Number.

Reinhard Zumkeller, Illustration of initial terms

Example

Divisors of n=18: {1,2,3,6,9,18}; 18 is pseudo-perfect (A005835): 18=9+6+3, but there exist no two complementary subsets of divisors having the same sum, therefore 18 is a term.

Mathematica

(* first do *) Needs["Combinatorica`"] (* then *) abQ[n_] := DivisorSigma[1, n] > 2 n; sq2sQ[n_] := IntegerQ@ Sqrt@ n || IntegerQ@ Sqrt@(n/2); fQ[n_] := Block[{d = Divisors@n, lmt = 1 + 2^DivisorSigma[0, n]/2, k, s}, k = 1 + Length@d; s = Plus @@ d/2; While[k < lmt && Plus @@ NthSubset[k, d] != s, k++ ]; If[k == lmt, True, False]]; lst = {}; k = 1; While[k < 10^3, If[abQ@k && sq2sQ@k && fQ@k, AppendTo[lst, k]; Print@k]; k++ ]; lst (* Robert G. Wilson v, Apr 01 2010 *)

Crossrefs

Disjoint union of A156903 and A171641. - Amiram Eldar, Jun 20 2020

Cf. A000203, A005101, A028982, A083206, A083210.

Sequence in context: A097926 A087967 A070224 * A156903 A204824 A252424

Adjacent sequences: A083208 A083209 A083210 * A083212 A083213 A083214

Keyword

nonn

Author

Reinhard Zumkeller, Apr 22 2003

Extensions

a(21)-a(46) from Robert G. Wilson v, Apr 01 2010

Status

approved