A347063 -id:A347063 - cp's OEIS frontend

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

Amiram Eldar, Oct 31 2021

nonn

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.

			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.

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

Cf. A027750, A057020, A057021, A083207.

A347063 is a subsequence.

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]

A364053 Zumkeller numbers whose divisors can be partitioned into two disjoint sets with equal products.

Original entry on oeis.org

6, 24, 30, 40, 42, 54, 56, 60, 66, 70, 78, 84, 88, 90, 96, 102, 104, 108, 114, 120, 126, 132, 138, 140, 150, 156, 160, 168, 174, 186, 198, 204, 210, 216, 220, 222, 224, 228, 234, 240, 246, 258, 260, 264, 270, 276, 280, 282, 294, 306, 308, 312, 318, 330, 336, 340, 342, 348, 350, 352

Offset: 1

Ivan N. Ianakiev, Jul 04 2023

nonn

Intersection of A083207 and A048943.

			The divisors of 24 are {1,2,3,4,6,8,12,24}. They can be partitioned into two disjoint sets with equal sums, namely {4,6,8,12} and {1,2,3,24}, and two disjoint sets with equal products, namely {1,2,12,24} and {3,4,6,8}. So, 24 is a term and also a term of A347063.

Cf. A083207, A048943, A347063.

zQ[n_]:=Module[{d=Divisors[n],t,ds,x},ds=Plus@@d;If[Mod[ds,2]>0,False,t=CoefficientList[Product[1+x^i,{i,d}],x];t[[1+ds/2]]>0]]; fQ[n_]:=IntegerQ[Sqrt[Times@@Divisors[n]]];
Select[Range[1000],And[fQ[#],zQ[#]]&] (*zQ and fQ by T. D. Noe at A083207 and A048943*)

A376862 Unitary Zumkeller numbers whose divisors can be partitioned into two disjoint subsets with equal sums and cardinalities.

Original entry on oeis.org

30, 42, 60, 66, 78, 90, 102, 114, 138, 150, 174, 186, 210, 222, 246, 258, 282, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690

Offset: 1

Ivan N. Ianakiev, Oct 07 2024

A unitary divisor of n is a divisor d such that gcd(d,n/d)=1.
This sequence is an intersection of A290466 and A347063 and seemingly a subsequence of A293188.
From the facts: a) for n>2 every primorial(n), i.e. A002110(n), is a Zumkeller number, b) a(1) = 30 = 2*3*5 is primorial(3), c) if n is squarefree, than sigma(n) = usigma(n), d) the number of unitary divisors of n is 2^k, where k is the number of distinct prime factors of n, and e) p*y belongs to A347063, where p is a prime coprime to y and y belongs to A347063, it follows that the present sequence is infinite, since for m >= 3 primorial(m) is a term.
It seems that for k >= 0 all numbers of the form 30 + 36k are terms.

			The set of divisors of 90 is {1,2,3,5,6,9,10,15,18,30,45,90}, which is a union of the sets {1,2,3,6,15,90} and {5,9,10,18,30,45}, which have equal sums (117) and cardinalities (6). So, 90 is a term.

Eric Weisstein's World of Mathematics, Unitary Divisor Function
Wikipedia, Unitary divisor

Cf. A083207, A290466, A347063.

uzn=Cases[Import["https://oeis.org/A290466/b290466.txt","Table"],{,}][[All,2]];
dzn=Select[Range@700,!IntegerQ@Sqrt@#&&(d=Divisors@#; MemberQ[Total/@Subsets[d,{Length@d/2}],Total@d/2])&]; Intersection[uzn,dzn] (* Thanks to Giorgos Kalogeropoulos at A347063 *)

A380289 Unitary Double Zumkeller numbers: numbers whose set of unitary divisors can be partitioned into two disjoint sets with equal sums and equal cardinalities.

Original entry on oeis.org

30, 42, 66, 78, 102, 114, 138, 150, 174, 186, 210, 222, 246, 258, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822, 834, 858, 870, 894, 906, 910, 930, 942, 966, 978, 990

Offset: 1

Ivan N. Ianakiev, Jan 19 2025

nonn

Based on checking the first 151 terms of this sequence it seems it is a subsequence of A342398. The first number that belongs to A342398, but not to this sequence is 2394. It also seems a subsequence of Zumkeller numbers (A083207). It is not a subsequence of Sphenic numbers (A007304). For example, 150 = 2*3*5*5 does not belong to A007304.
If y is a term, then so is x*y, where x is coprime to y.
It seems that 12k+6 is a term, where k>0 and k == 0 or 2 mod 3. Verified for k <= 191.

			Let D be the set of unitary divisors of 210. D = {1,2,3,5,6,7,10,14,15,21,30,35,42,70,105,210} = {1,2,5,6,14,15,35,210}union{3,7,10,21,30,42,70,105}.

Bhabesh Das, On unitary Zumkeller numbers, Notes on Number Theory and Discrete Mathematics, Vol. 30, No. 2 (2024), pp. 436-442.
Eric Weisstein's World of Mathematics, Unitary Divisor Function.
Wikipedia, Unitary divisor.

Cf. A083207, A290466 (supersequence), A342398, A347063.

uDiv[n_]:=Block[{d=Divisors[n]},Select[d,GCD[#,n/#]==1&]];uZNQ[n_]:=Module[{d=uDiv[n],t,ds,x},ds=Plus@@d;If[Mod[ds,2]>0,False,t=CoefficientList[Product[1+x^i,{i,d}],x];t[[1+ds/2]]>0]];dZNQ[n_]:=Block[{div=uDiv[n]},!IntegerQ@Sqrt[n]&&MemberQ[Total/@Subsets[div,{Length@div/2}],Total@div/2]];Select[Range[1000],uZNQ[#]&&dZNQ[#]&]

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.

Original entry on oeis.org

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

Ivan N. Ianakiev, Mar 09 2025

nonn

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.

			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.

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

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

cp's OEIS Frontend

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

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A364053 Zumkeller numbers whose divisors can be partitioned into two disjoint sets with equal products.

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A376862 Unitary Zumkeller numbers whose divisors can be partitioned into two disjoint subsets with equal sums and cardinalities.

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A380289 Unitary Double Zumkeller numbers: numbers whose set of unitary divisors can be partitioned into two disjoint sets with equal sums and equal cardinalities.

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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.

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