A381874 Numbers whose set of divisors can be partitioned: a) into two disjoint subsets with equal sums and cardinalities, and b) into two disjoint subsets with equal products and cardinalities.

24, 30, 42, 54, 60, 66, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 140, 150, 156, 160, 168, 174, 186, 198, 204, 210, 216, 220, 222, 224, 228, 240, 246, 258, 260, 264, 270, 276, 280, 282, 306, 308, 312, 318, 330, 336, 340, 342, 348, 352, 354, 360, 364, 366, 372, 378, 380, 384, 390, 402

Offset: 1

Author

Ivan N. Ianakiev, Mar 09 2025

Keywords

• nonn

Comments

If y is a term of this sequence, then so is p*y, where p is a prime that is coprime to y.

It seems that the maximum first difference is 24.

Examples

			Let D be the set of the divisors of 24, i.e. {1,2,3,4,6,8,12,24}. On the one hand (sums) D = {1,2,3,24} union {4,6,8,12}. On the other hand (products) D = {1,2,12,24} union {3,4,6,8}. So, 24 is a term.
		

Crossrefs

Cf. A083207 (supersequence), A347063 (supersequence).

Programs

Mathematica

sumQ[set_]:=AnyTrue[Subsets[set,{Length[set]/2}],(Plus@@#==Total[Complement[set,#]]&)];
prodQ[set_]:=AnyTrue[Subsets[set,{Length[set]/2}],(Times@@#==Times@@Complement[set,#]&)];
Select[Range[500],sumQ[Divisors[#]]&&prodQ[Divisors[#]]&]//Quiet