A290466 -id:A290466 - cp's OEIS frontend

A348527 Noninfinitary Zumkeller numbers: numbers whose set of noninfinitary divisors is nonempty and can be partitioned into two disjoint sets of equal sum.

48, 80, 96, 112, 150, 180, 240, 252, 294, 336, 360, 396, 432, 468, 480, 486, 504, 528, 560, 600, 612, 624, 630, 672, 684, 720, 726, 768, 792, 810, 816, 828, 864, 880, 912, 936, 960, 1008, 1014, 1040, 1044, 1050, 1056, 1104, 1116, 1120, 1134, 1176, 1200, 1232, 1248

Offset: 1

Amiram Eldar, Oct 21 2021

nonn

The smallest odd term is a(104) = 2475.

			48 is a term since its set of noninfinitary divisors, {2, 4, 6, 8, 12, 24}, can be partitioned into the two disjoint sets, {2, 6, 8, 12} and {4, 24}, whose sums are equal: 2 + 6 + 8 + 12 = 4 + 24 = 28.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A348271, A348274, A348341.

Similar sequences: A083207, A290466, A335215, A335142, A335197, A335218.

nidiv[1] = {}; nidiv[n_] := Complement[Divisors[n], Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; nizQ[n_] := Module[{d = nidiv[n], sum, x}, sum = Plus @@ d; sum > 0 && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; Select[Range[1250], !IntegerQ@ Log2@ DivisorSigma[0, #] && nizQ[#] &]

A384515 Unitary s-Zumkeller numbers.

Original entry on oeis.org

60, 140, 420, 660, 1224, 1820, 2660, 2820, 4620, 5460, 7140, 7980, 8580, 9660, 11220, 12180, 12540, 13020, 13260, 13580, 13860, 14140, 14420, 14820, 15180, 15540, 16380, 17220, 17940, 18060, 18200, 19140, 19380, 19740, 20020, 20460, 22260, 22620, 23460, 24180, 24420

Offset: 1

Ivan N. Ianakiev, Jun 01 2025

nonn

A positive integer k is called a unitary s-Zumkeller number if the set D of proper positive unitary divisors of k can be partitioned as {A,B}, such that the sum of the squares of the elements of A equals the sum of the squares of the elements of B equals (sigma2U(k) - k^2)/2, where sigma2U(k) is the sum of the squares of the unitary divisors of k.

All terms are even numbers.

			The set of proper unitary divisors of 60 is A = {1,3,4,5,12,15,20}. The set of their squares is B= {1,9,16,25,144,225,400} and the sum of its elements is 820. B = {1,9,400} union {16,25,144,225}, which makes 60 a term of the present sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI program.
Jayanta Kalita and Helen K. Saikia, A Study On Generalized Zumkeller Numbers, INTEGERS, Volume 24 (2024), A95, pp. 7-10.

Cf. A034448, A077610, A083207, A290466.

(* Naive code to illustrate the terms found by Kalita and Saikia *)
uDiv[n_]:=Block[{d=Divisors[n]},Select[d,GCD[#,n/#]==1&]];
sigma2UDiv[n_]:=Total[uDiv[n]^2]; propUDiv[n_]:=uDiv[n]//Most;
subsetsPropUDivQ[1]:=False; subsetsPropUDivQ[n_]:=
Select[Subsets[propUDiv[n]],Total[Flatten[#]^2]==(sigma2UDiv[n]-n^2)/2&]!={};
Select[Range[1820],subsetsPropUDivQ[#]&]
(* or *)
q[n_] := Module[{d = Select[Divisors[n], # < n && CoprimeQ[#, n/#] &]^2, sum, x}, sum = Total[d]; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; Select[Range[2, 1000], q] (* Amiram Eldar, Jun 01 2025 *)

PARI
```
\\ See Corneth link
```

a(7)-a(16) from Amiram Eldar, Jun 01 2025

More terms from David A. Corneth, Jun 01 2025

A335145 Numbers that are both unitary and nonunitary Zumkeller numbers.

Original entry on oeis.org

150, 294, 630, 726, 750, 840, 1014, 1050, 1470, 1650, 1734, 1890, 1950, 2058, 2166, 2550, 2850, 2940, 2970, 3174, 3234, 3450, 3510, 3630, 3750, 3822, 4350, 4410, 4650, 4998, 5046, 5070, 5082, 5250, 5550, 5586, 5670, 5766, 6150, 6450, 6762, 6930, 7050, 7098, 7260

Offset: 1

Amiram Eldar, May 25 2020

nonn

			150 is a term since its unitary divisors, {1, 2, 3, 6, 25, 50, 75, 150} can be partitioned in two disjoint sets of equal sum: 1 + 2 + 3 + 25 + 50 + 75 = 6 + 150, and so are its nonunitary divisors, {5, 10, 15, 30}: 5 + 10 + 15 = 30.

Intersection of A290466 and A335142.

zumQ[n_] := Module[{d = Divisors[n], ud, nd, sumUd, sumNd, x},ud = Select[d, CoprimeQ[#, n/#] &]; nd = Complement[d, ud]; sumUd = Plus @@ ud; sumNd = Plus @@ nd; sumUd >= 2*n && sumNd > 0 && EvenQ[sumUd] && EvenQ[sumNd] && CoefficientList[Product[1 + x^i, {i, ud}], x][[1 + sumUd/2]] > 0 && CoefficientList[Product[1 + x^i, {i, nd}], x][[1 + sumNd/2]] > 0]; Select[Range[10000], zumQ]

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[#]&]

cp's OEIS Frontend

A348527 Noninfinitary Zumkeller numbers: numbers whose set of noninfinitary divisors is nonempty and can be partitioned into two disjoint sets of equal sum.

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A384515 Unitary s-Zumkeller numbers.

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PARI

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A335145 Numbers that are both unitary and nonunitary Zumkeller numbers.

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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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