A293188 -id:A293188 - cp's OEIS frontend

A342119 Numbers k with property that if k has m divisors, there are m/2 divisors of k whose sum is k.

2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 23, 24, 29, 30, 31, 37, 40, 41, 42, 43, 47, 48, 53, 54, 59, 60, 61, 67, 71, 72, 73, 79, 80, 83, 84, 89, 90, 96, 97, 101, 103, 107, 108, 109, 112, 113, 120, 126, 127, 131, 132, 137, 139, 140, 149, 150, 151, 156, 157, 160, 162

Offset: 1

Metin Sariyar, Feb 28 2021

nonn

All primes and all numbers of the form 3*2^k (k>1) are terms.

			40 is a term because it has 8 divisors and 2+8+10+20 = 40.

Cf. A000203, A293188, A005835.

Select[Range[160], EvenQ[(d = DivisorSigma[0, #])] && MemberQ[Plus @@@ Subsets[Divisors[#], {d/2}], #] &] (* Amiram Eldar, Feb 28 2021 *)

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.

Original entry on oeis.org

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]

A349542 Number of partitions of n into distinct unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 0

Ilya Gutkovskiy, Nov 21 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 0..20000

Cf. A033630, A066874, A077610, A286851, A293188.

a[n_] := SeriesCoefficient[Product[(1 + Boole[GCD[n/d, d] == 1] x^d), {d, Divisors[n]}], {x, 0, n}]; Table[a[n], {n, 0, 114}]

A349542(n) = if(!n, 1, my(p=1); fordiv(n,d, if(1==gcd(d,n/d), p *= (1 + 'x^d))); polcoeff(p,n)); \\ Antti Karttunen, Nov 22 2021

a(n) = [x^n] Product_{d|n, gcd(d,n/d) = 1} (1 + x^d).

A342402 Numbers k with property that if k has m proper divisors, there are m/2 proper divisors of k whose sum is k.

Original entry on oeis.org

36, 100, 144, 324, 400, 576, 784, 900, 1296, 1600, 1764, 1936, 2304, 2500, 2704, 2916, 3136, 3600, 4356, 4624, 4900, 5184, 5776, 6084, 6400, 7056, 7744, 8100, 9216, 9604, 10000, 10404, 10816, 11025, 11664, 12100, 12544, 12996, 14400, 15876, 16900, 17424, 18496

Offset: 1

Metin Sariyar, Mar 10 2021

nonn

All terms are perfect squares.

			576 is a term because it has 20 proper divisors and 1+2+3+4+6+8+24+48+192+288 = 576.

Cf. A000396, A001065, A005835, A293188, A342119.

q[n_] := Module[{d = Most @ Divisors[n], nd, ans = False}, nd = Length[d]; Do[If[Plus @@ Subsets[d, {nd/2}, {k}][[1]] == n, ans = True; Break[]], {k, 1, Binomial[nd, nd/2]}]; ans]; Select[Range[136]^2, q] (* Amiram Eldar, Mar 11 2021 *)

a(8)-a(43) from Alois P. Heinz, Mar 10 2021

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 *)

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A342119 Numbers k with property that if k has m divisors, there are m/2 divisors of k whose sum is k.

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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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A349542 Number of partitions of n into distinct unitary divisors of n.

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A342402 Numbers k with property that if k has m proper divisors, there are m/2 proper divisors of k whose sum is k.

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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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