A348713 Numbers whose divisors can be partitioned into two disjoint sets with equal arithmetic mean.

6, 20, 24, 30, 42, 48, 54, 56, 60, 66, 70, 72, 78, 84, 88, 90, 96, 102, 108, 114, 120, 126, 132, 135, 138, 140, 150, 156, 160, 168, 174, 180, 186, 190, 192, 196, 198, 200, 204, 210, 216, 220, 222, 224, 228, 230, 234, 240, 246, 252, 258, 260, 264, 270, 273, 276

Offset

1,1

Comments

The arithmetic mean of each of the two subsets is equal to the arithmetic mean of all the divisors of the number.

Also, numbers whose divisors can be partitioned into two disjoint sets with equal harmonic mean. This definition is equivalent since the harmonic mean of a subset {d_i} of the divisors of k is equal to k/<k/d_i>, where <k/d_i> is the arithmetic mean over the complementary divisors k/d_i.

Links

Amiram Eldar, Table of n, a(n) for n = 1..872

Example

6 is a term since its set of divisors, {1, 2, 3, 6}, can be partitioned into the two disjoint sets, {3} and {1, 2, 6}, whose arithmetic means are both 3.

Mathematica

q[n_] := Module[{d = Divisors[n], nd, m, s, subs, ans = False}, nd = Length[d]; m = Plus @@ d/nd; subs = Subsets[d]; Do[s = subs[[k]]; If[0 < Length[s] < nd && Mean[s] == m, ans = True; Break[]], {k, 1, Length[subs]}]; ans]; Select[Range[300], q]

Crossrefs

Cf. A027750, A057020, A057021, A083207.

A347063 is a subsequence.

Sequence in context: A243905 A062017 A103678 * A020889 A334817 A084682

Adjacent sequences: A348710 A348711 A348712 * A348714 A348715 A348716

Keyword

nonn

Author

Amiram Eldar, Oct 31 2021

Status

approved