

A083211


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


4



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
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OFFSET

1,1


COMMENTS

A083206(a(n))=0; subsequence of A083210.
From Robert G. Wilson v, Apr 01 2010: (Start)
All members must be even because if odd then the two subsets will be of opposite parity.
All members must be either a square or twice a square (A028982).
If k is present the so is 2k. Therefore the primitive subset is 18, 100, 162, 196, 450, 748, 774, 846, 882, 954, 968, 1062, 1098, ..., .
Most of the members have odd abundances (A156903), but there are exceptions: 738, 846, 954, 1062, 1098, ..., . (End)


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 pseudoperfect (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

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



