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The sum of the proper divisors of n, weighted by divisor multiplicity, equals n.

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`%I #6 May 25 2019 22:08:26
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`%S 6,152,656,2888,18632,36224,55328384,1082574464
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`%N The sum of the proper divisors of n, weighted by divisor multiplicity, equals n.
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`%C The multiplicity of a divisor d > 1 in n is defined as the largest power i for which d^i divides n; otherwise is defined as 1 if d = 1.
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`%C From _Ray Chandler_, Dec 08 2009: (Start)
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`%C Also in the sequence, but not necessarily the next terms,
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`%C 2^k * p where p = A168512(2^k) is prime:
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`%C 2^18 * 525529 = 137764274176,
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`%C 2^25 * 67117859 = 2252101635801088,
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`%C 2^26 * 134234921 = 9008353057439744,
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`%C 2^30 * 2147551801 = 2305916187940225024,
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`%C 2^40 * 2199025372073 = 2417853966368708281499648,
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`%C 2^50 * 2251799880936649 = 2535301276174804923929356926976,
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`%C as well as k = 150, 348, 694, ... (End)
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`%e The proper divisors of 152 are 1, 2, 4, 8, 19, 38, 76 of multiplicity 1, 3, 1, 1, 1, 1, 1 respectively. Since 1*1 + 3*2 + 1*4 + 1* 8 + 1*19 + 1*38 + 1*76 = 152, then 152 belongs to the sequence.
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`%t (*multiplicity of d in n*) divmult[d_, n_] := Module[{output, i}, If[d == 1, output = 1, If[d == n, output = 1, i = 0; While[Mod[n, d^(i + 1)] == 0, i = i + 1]; output = i]]; output]; (*sum of divisors weighted by divisor multiplicity*) dmt[n_] := Module[{divs, l}, divs = Divisors[n]; l = Length[divs]; Sum[divmult[divs[[i]], n]*divs[[i]], {i, 1, l}]]; (*search for sequence terms*) ls = {}; Do[If[dmt[n] == 2 n, ls = Append[ls, n]], {n, 2, 10^7}]; ls
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`%Y Cf. A168512
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`%K more,nonn
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`%O 1,1
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`%A _Joseph L. Pe_, Dec 01 2009
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`%E a(7)-a(8) from _Ray Chandler_, Dec 08 2009
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