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

Author

Amiram Eldar, Oct 31 2021

Keywords

• nonn

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/, where is the arithmetic mean over the complementary divisors k/d_i.

Examples

			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.
		

Links

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

Crossrefs

Cf. A027750, A057020, A057021, A083207.

A347063 is a subsequence.

Programs

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]