A342398 -id:A342398 - cp's OEIS frontend

A342399 Unitary pseudoperfect numbers k such that no subset of the nontrivial unitary divisors {d|k : 1 < d < k, gcd(d, k/d) = 1} adds up to k.

3510, 3770, 5670, 5810, 6790, 7630, 7910, 9590, 9730, 544310, 740870, 2070970, 4017310, 4095190, 5368510, 5569690, 5762330, 5838770, 5855290, 5856130, 5887630, 5902470, 5985770, 6006070, 6039530, 6075370, 6083630, 6181210, 6259610, 6471290, 7038710, 7065730, 7285390

Offset: 1

Amiram Eldar, Mar 10 2021

nonn

Numbers that are the sum of a proper subset of their aliquot unitary divisors but are not the sum of any subset of their nontrivial unitary divisors.
The unitary perfect numbers (A002827) which are a subset of the unitary pseudoperfect numbers (A293188) are excluded from this sequence since otherwise they would all be trivial terms: if k is a unitary perfect number then the sum of the divisors {d|k : 1 < d < k, gcd(d, k/d) = 1} is k-1, so any subset of them has a sum smaller than k.
The unitary pseudoperfect numbers are thus a disjoint union of the unitary perfect numbers, this sequence and A342398.
The unitary abundant numbers (A034683) are a disjoint union of the unitary weird numbers (A064114), this sequence and A342398.

			3510 is a term since it is a unitary pseudoperfect number, 3510 = 1 + 2 + 5 + 13 + 27 + 54 + 65 + 130 + 135 + 270 + 351 + 702 + 1755, and the set of nontrivial unitary divisors of 3510, {d|3510 : 1 < d < 3510, gcd(d, 3510/d) = 1} = {2, 5, 10, 13, 26, 27, 54, 65, 130, 135, 270, 351, 702, 1755}, has no subset that adds up to 3510.

Amiram Eldar, Table of n, a(n) for n = 1..206

The unitary version of A339343.
Subsequence of A034683 and A293188.
Cf. A002827, A064114, A342398.

q[n_] := Module[{d = Most @ Select[Divisors[n], CoprimeQ[#, n/#] &], x}, Plus @@ d > n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0 && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, 2, Length[d]}], {x, 0, n}], n] == 0]; Select[Range[10^4], q]

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

A342399 Unitary pseudoperfect numbers k such that no subset of the nontrivial unitary divisors {d|k : 1 < d < k, gcd(d, k/d) = 1} adds up to k.

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