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A083211
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Abundant numbers (A005101) with no subset of their divisors such that the complement has the same sum.
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4
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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
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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