login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Numbers whose divisors can be partitioned into two disjoint sets without singletons whose arithmetic means are both integers.
2

%I #13 Nov 03 2021 12:20:02

%S 6,10,12,14,15,16,18,20,21,22,24,26,27,28,30,32,33,34,35,36,38,39,40,

%T 42,44,45,46,48,50,51,52,54,55,56,57,58,60,62,63,64,65,66,68,69,70,72,

%U 74,75,76,77,78,80,81,82,84,85,86,87,88,90,91,92,93,94,95,96

%N Numbers whose divisors can be partitioned into two disjoint sets without singletons whose arithmetic means are both integers.

%C First differs from A343311 at n = 29.

%C Differs from A080257 which contains for example 8 and 128. - _R. J. Mathar_, Nov 03 2021

%H Amiram Eldar, <a href="/A348718/b348718.txt">Table of n, a(n) for n = 1..1000</a>

%e 6 is a term since its set of divisors, {1, 2, 3, 6}, can be partitioned into the two disjoint sets {1, 3} and {2, 6} whose arithmetic means, 2 and 4 respectively, are both integers.

%t amQ[d_] := IntegerQ @ Mean[d]; q[n_] := Module[{d = Divisors[n], nd, s, subs, ans = False}, nd = Length[d]; subs = Subsets[d]; Do[s = subs[[k]]; If[Length[s] > 1 && Length[s] <= nd/2 && amQ[s] && amQ[Complement[d, s]], ans = True; Break[]], {k, 1, Length[subs]}]; ans]; Select[Range[100], q]

%Y Cf. A003601, A027750, A057020, A057021, A083207, A343311, A348715.

%K nonn

%O 1,1

%A _Amiram Eldar_, Oct 31 2021