A290466 -id:A290466 - cp's OEIS frontend

A335202 Unitary Zumkeller numbers (A290466) whose set of unitary divisors can be partitioned into two disjoint sets of equal sum in a single way.

6, 60, 70, 90, 3230, 3770, 4030, 4510, 5170, 5390, 5830, 50388, 87360, 269990, 442365, 544310, 592670, 740870, 1341230, 1772870, 4173070, 4199030, 5719266, 5728842, 5743206, 34473582, 624032630, 812851182, 1109686930, 1113445430, 2280959890, 55157757606

Offset: 1

Amiram Eldar, May 26 2020

nonn

			60 is a term since there is only one partition of its set of unitary divisors, {1, 3, 4, 5, 12, 15, 20, 60}, into 2 disjoint sets whose sum is equal: 1 + 3 + 4 + 5 + 12 + 15 + 20 = 60.

Giovanni Resta, Table of n, a(n) for n = 1..48 (terms < 10^12)

The unitary version of A083209.
Subsequence of A290466.
A002827 is a subsequence.

uzQ[n_] := Module[{d = Select[Divisors[n], CoprimeQ[#, n/#] &], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] == 2]]; Select[Range[6000], uzQ]

Terms a(19) and beyond from Giovanni Resta, May 30 2020

A335201 Unitary Zumkeller numbers (A290466) that are not squarefree.

Original entry on oeis.org

60, 90, 150, 294, 420, 630, 660, 726, 750, 780, 840, 924, 990, 1014, 1020, 1050, 1092, 1140, 1170, 1380, 1386, 1428, 1470, 1530, 1596, 1638, 1650, 1710, 1734, 1740, 1860, 1890, 1950, 2058, 2070, 2142, 2166, 2220, 2460, 2550, 2580, 2610, 2790, 2820, 2850, 2940

Offset: 1

Amiram Eldar, May 26 2020

nonn

Zumkeller numbers (A083207) that are squarefree (A005117) are also unitary Zumkeller numbers (A290466), since all of their divisors are unitary.

First differs from A335140 at n = 39.

			60 is a term since it is nonsquarefree, and its unitary divisors, {1, 3, 4, 5, 12, 15, 20, 60}, can be partitioned into 2 disjoint sets whose sum is equal: 1 + 3 + 4 + 5 + 12 + 15 + 20 = 60.

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

Intersection of A013929 and A290466.

Cf. A005117, A083207.

uzQ[n_] :=  Module[{d = Select[Divisors[n], CoprimeQ[#, n/#] &], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]]; Select[Range[3000], !SquareFreeQ[#] && uzQ[#] &]

A002827 Unitary perfect numbers: numbers k such that usigma(k) - k = k.

Original entry on oeis.org

6, 60, 90, 87360, 146361946186458562560000

Offset: 1

N. J. A. Sloane

d is a unitary divisor of k if gcd(d,k/d)=1; usigma(k) is their sum (A034448).
The prime factors of a unitary perfect number (A002827) are the Higgs primes (A057447). - Paul Muljadi, Oct 10 2005
It is not known if a(6) exists. - N. J. A. Sloane, Jul 27 2015
Frei proved that if there is a unitary perfect number that is not divisible by 3, then it is divisible by 2^m with m >= 144, it has at least 144 distinct odd prime factors, and it is larger than 10^440. - Amiram Eldar, Mar 05 2019
Conjecture: Subsequence of A083207 (Zumkeller numbers). Verified for all present terms. - Ivan N. Ianakiev, Jan 20 2020

			Unitary divisors of 60 are 1,4,3,5,12,20,15,60, with sum 120 = 2*60.
146361946186458562560000 = 2^18 * 3 * 5^4 * 7 * 11 * 13 * 19 * 37 * 79 * 109 * 157 * 313.

R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.
F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, p. 59, 1983.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.45.1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 147-148.

H. A. M. Frei, Über unitar perfekte Zahlen, Elemente der Mathematik, Vol. 33, No. 4 (1978), pp. 95-96.
Takeshi Goto, Upper Bounds for Unitary Perfect Numbers and Unitary Harmonic Numbers, Rocky Mountain Journal of Mathematics, Vol. 37, No. 5 (2007), pp. 1557-1576.
A. V. Lelechenko, The Quest for the Generalized Perfect Numbers, in Theoretical and Applied Aspects of Cybernetics, TAAC 2014, Kiev.
M. V. Subbarao, Letter to N. J. A. Sloane, Feb 18 1974
M. V. Subbarao, T. J. Cook, R. S. Newberry and J. M. Weber, On unitary perfect numbers, Delta, 3 (No. 1, 1972), 22-26.
G. Villemin's Almanac of Numbers, Nombres Unitairement Parfaits
C. R. Wall, Letter to P. Hagis, Jr., Jan 13 1972
C. R. Wall, The fifth unitary perfect number, Canad. Math. Bull., 18 (1975), 115-122.
C. R. Wall, On the largest odd component of a unitary perfect number, Fib. Quart., 25 (1987), 312-316.
Eric Weisstein's World of Mathematics, Unitary Perfect Number.
Wikipedia, Unitary perfect number

Cf. A034460, A034448, A057447.
Subsequence of the following sequences: A003062, A290466 (seemingly), A293188, A327157, A327158.
Gives the positions of ones in A327159.

usnQ[n_]:=Total[Select[Divisors[n],GCD[#,n/#]==1&]]==2n; Select[Range[ 90000],usnQ] (* This will generate the first four terms of the sequence; it would take a very long time to attempt to generate the fifth term. *) (* Harvey P. Dale, Nov 14 2012 *)

is(n)=sumdivmult(n, d, if(gcd(d, n/d)==1, d))==2*n \\ Charles R Greathouse IV, Aug 01 2016

If m is a term and omega(m) = A001221(m) = k, then m < 2^(2^k) (Goto, 2007). - Amiram Eldar, Jun 06 2020

A328328 Unitary admirable numbers: numbers k such that there is a proper unitary divisor d of k such that usigma(k) - 2d = 2k, where usigma is the sum of unitary divisors function (A034448).

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 114, 138, 150, 174, 186, 222, 246, 258, 282, 294, 318, 354, 366, 402, 420, 426, 438, 474, 498, 534, 582, 606, 618, 630, 642, 654, 660, 678, 726, 750, 762, 780, 786, 822, 834, 840, 894, 906, 942, 978, 990, 1002, 1014, 1020, 1038, 1074, 1086

Offset: 1

Amiram Eldar, Oct 12 2019

nonn

Differs from A302574(n) at n >= 30.
Equivalently, numbers that equal to the sum of their proper unitary divisors, with one of them taken with a minus sign.
The unitary version of A111592.
The squarefree terms are also admirable numbers (A111592). The nonsquarefree terms are 150, 294, 420, 630, 660, 726, 750, 780, 840, 990, ...
The unitary abundant numbers (A034683) that are not unitary admirable numbers are: 210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858, 870, 910, 924, 930, 966, ...

			150 is in the sequence since 150 = 1 + 2 + 3 - 6 + 25 + 50 + 75 is the sum of its proper unitary divisors with one of them, 6, taken with a minus sign.

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

Cf. A034448, A077610, A111592, A302574.

Subsequence of A034683 and A290466.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); aQ[n_] := (ab = usigma[n] - 2n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && CoprimeQ[2*n/ab, ab/2]; Select[Range[1086], aQ]

A323341 Numbers k whose unitary divisors have an even sum which is larger than 2k, but they cannot be partitioned into two disjoint parts whose sums are equal.

Original entry on oeis.org

2394, 1452330, 5771934, 5786298, 5800662, 5834178, 5843754, 5858118, 5886846, 5905998, 5920362, 5929938, 5992182, 6035274, 6059214, 6078366, 6087942, 6102306, 6107094, 6121458, 6174126, 6202854, 6207642, 6245946, 6265098, 6274674, 6303402, 6336918, 6360858

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

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

The unitary version of A171641.

Cf. A034448, A034683, A290466, A323342, A323343, A323344.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; seq={}; Do[s=usigma[n]; If[OddQ[s] || s<=2n, Continue[]]; udiv = Select[Divisors[n], GCD[ #, n/# ] == 1 &]; If[Coefficient[Times @@ (1 + x^udiv) // Expand, x, s/2] == 0, AppendTo[seq, n]], {n, 1, 1500000}]; seq

A335215 Bi-unitary Zumkeller numbers: numbers whose set of bi-unitary divisors can be partitioned into two disjoint sets of equal sum.

Original entry on oeis.org

6, 24, 30, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 88, 90, 96, 102, 104, 114, 120, 138, 150, 160, 162, 168, 174, 186, 192, 210, 216, 222, 224, 240, 246, 258, 264, 270, 280, 282, 288, 294, 312, 318, 320, 330, 336, 352, 354, 360, 366, 378, 384, 390, 402

Offset: 1

Amiram Eldar, May 27 2020

nonn

			6 is a term since its set of bi-unitary divisors, {1, 2, 3, 6}, can be partitioned into 2 disjoint sets, whose sum is equal: 1 + 2 + 3 = 6.

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

The bi-unitary version of A083207.
Subsequence of A292982.
Cf. A188999, A222266, A290466, A323342, A335142, A335197.

uDivs[n_] := Select[Divisors[n], CoprimeQ[#, n/#] &]; bDivs[n_] := Select[Divisors[n], Last @ Intersection[uDivs[#], uDivs[n/#]] == 1 &]; bzQ[n_] := Module[{d = bDivs[n], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]]; Select[Range[10^3], bzQ]

A339979 Coreful Zumkeller numbers: numbers whose set of coreful divisors can be partitioned into two disjoint sets of equal sum.

Original entry on oeis.org

36, 72, 144, 180, 200, 252, 288, 324, 360, 392, 396, 400, 468, 504, 576, 600, 612, 648, 684, 720, 784, 792, 800, 828, 900, 936, 1008, 1044, 1116, 1152, 1176, 1200, 1224, 1260, 1296, 1332, 1368, 1400, 1440, 1476, 1548, 1568, 1584, 1600, 1620, 1656, 1692, 1764

Offset: 1

Amiram Eldar, Dec 25 2020

nonn

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

The coreful perfect numbers (A307958) are a subsequence.

			36 is a term since its set of coreful divisors, {6, 12, 18, 36}, can be partitioned into the two disjoint sets, {6, 12, 18} and {36}, whose sums are equal: 6 + 12 + 18 = 36.

A307958 is a subsequence.
Subsequence of A308053.
Cf. A007947, A057723.
Similar sequences: A083207, A290466, A335197, A335142, A335215, A335218.

corZumQ[n_] := Module[{r = Times @@ FactorInteger[n][[;; , 1]], d, sum, x}, d = r * Divisors[n/r]; (sum = Plus @@ d) >= 2*n && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; Select[Range[1800], corZumQ]

from itertools import count, islice
from sympy import primefactors, divisors
def A339979_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = primefactors(n)
        d = [x for x in divisors(n) if primefactors(x)==f]
        s = sum(d)
        if s&1^1 and n<<1<=s:
            d = d[:-1]
            s2, ld = (s>>1)-n, len(d)
            z = [[0 for  in range(s2+1)] for  in range(ld+1)]
            for i in range(1, ld+1):
                y = min(d[i-1], s2+1)
                z[i][:y] = z[i-1][:y]
                for j in range(y,s2+1):
                    z[i][j] = max(z[i-1][j],z[i-1][j-y]+y)
                if z[i][s2] == s2:
                    yield n
                    break
A339979_list = list(islice(A339979_gen(),20)) # Chai Wah Wu, Feb 14 2023

A290467 Unitary half-Zumkeller numbers: numbers k whose unitary proper divisors can be partitioned into two disjoint sets whose sums are equal.

Original entry on oeis.org

6, 12, 20, 30, 42, 56, 60, 66, 70, 72, 78, 84, 90, 102, 114, 120, 138, 150, 168, 174, 180, 186, 210, 220, 222, 240, 246, 252, 258, 272, 280, 282, 294, 318, 330, 354, 360, 364, 366, 390, 402, 420, 426, 438, 440, 462, 474, 498, 510, 520, 532, 534, 546, 560, 570, 582, 606, 618

Offset: 1

Ivan N. Ianakiev, Aug 03 2017

nonn

Unitary divisors of n are divisors d such that gcd(d,n/d)=1.
Seemingly, a subsequence of A246198 (half-Zumkeller numbers).
The conjecture above is false, since 72, 3600 and 19600 do not belong to A246198. - Ivan N. Ianakiev, Jan 08 2025

			The set of unitary proper divisors of 12 is {1,3,4}. It can be partitioned into two disjoint subsets with equal sums of elements: {1,3} and {4}, therefore 12 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
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, A246198, A290466.

uPropDiv[n_/;n>1]:=Block[{d=Most[Divisors[n]]},Select[d,GCD[#,n/#]==1&]];uhZNQ[n_]:=Module[{d=uPropDiv[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]];Select[Range[10^3],uhZNQ] (* combined from the code by Robert G. Wilson v at A034448 and T. D. Noe at A083207 *)

A334408 Numbers k whose unitary divisors can be partitioned into two disjoint sets with equal sum, such that if d is in one set, then k/d is in the other set.

Original entry on oeis.org

462, 858, 870, 1482, 2310, 2730, 3570, 3990, 4002, 4290, 4620, 4830, 5460, 5610, 6006, 6090, 6270, 6438, 6510, 6630, 6930, 7140, 7410, 7770, 7854, 7998, 8190, 8580, 8610, 8778, 8970, 9240, 9570, 9660, 9870, 10010, 10230, 10374, 10626, 10920, 11220, 11310, 11550

Offset: 1

Amiram Eldar, Apr 27 2020

nonn

The squarefree terms of A334407 are also terms of this sequence. Terms that are not squarefree are 4620, 5460, 6930, 7140, 8190, 8580, 9240, 9660, ...

			462 is a term since its set of unitary divisors can be partitioned into two disjoint subsets: {1, 11, 14, 22, 66, 77, 154, 231} and {462, 42, 33, 21, 7, 6, 3, 2} = {462/1, 462/11, 462/14, 462/22, 462/66, 462/77, 462/154, 462/231} with the equal sum of 576, and with no pair of complementary unitary divisors (d, 462/d) in the same subset.

Amiram Eldar, Table of n, a(n) for n = 1..680

Subsequence of A290466.

Cf. A077610, A334407.

seqQ[n_] := Module[{d = Select[Divisors[n], CoprimeQ[#, n/#] &]}, nd = Length[d]; divpairs = d[[-1 ;; nd/2 + 1 ;; -1]] - d[[1 ;; nd/2]]; sd = Plus @@ divpairs; If[OddQ[sd], False, SeriesCoefficient[Series[Product[1 + x^divpairs[[i]], {i, Length[divpairs]}], {x, 0, sd/2}], sd/2] > 0]]; Select[Range[2, 10000], seqQ]

A335216 Bi-unitary Zumkeller numbers (A335215) that are not exponentially odd numbers (A268335).

Original entry on oeis.org

48, 60, 72, 80, 90, 150, 162, 192, 240, 288, 294, 320, 336, 360, 420, 432, 448, 504, 528, 540, 560, 576, 600, 624, 630, 648, 660, 720, 726, 756, 768, 780, 792, 800, 810, 816, 832, 880, 912, 924, 936, 960, 990, 1008, 1014, 1020, 1040, 1050, 1092, 1104, 1134, 1140

Offset: 1

Amiram Eldar, May 27 2020

nonn

Zumkeller numbers (A083207) that are exponentially odd (A268335) are also bi-unitary Zumkeller numbers (A335215), since all of their divisors are bi-unitary.

			48 is a term since it is not exponentially odd number (48 = 2^4 * 3 and 4 is even), and its set of bi-unitary divisors, {1, 2, 3, 6, 8, 16, 24, 48}, can be partitioned into 2 disjoint sets, whose sum is equal: 1 + 2 + 3 + 8 + 16 + 24 = 6 + 48.

Subsequence of A335215.

Cf. A083207, A268335, A290466.

uDivs[n_] := Select[Divisors[n], CoprimeQ[#, n/#] &]; bDivs[n_] := Select[Divisors[n], Last @ Intersection[uDivs[#], uDivs[n/#]] == 1 &]; bzQ[n_] := Module[{d = bDivs[n], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]]; expOddQ[n_] := AllTrue[Last /@ FactorInteger[n], OddQ]; Select[Range[1000], !expOddQ[#] && bzQ[#] &]

cp's OEIS Frontend

A335202 Unitary Zumkeller numbers (A290466) whose set of unitary divisors can be partitioned into two disjoint sets of equal sum in a single way.

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A335201 Unitary Zumkeller numbers (A290466) that are not squarefree.

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A002827 Unitary perfect numbers: numbers k such that usigma(k) - k = k.

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A328328 Unitary admirable numbers: numbers k such that there is a proper unitary divisor d of k such that usigma(k) - 2d = 2k, where usigma is the sum of unitary divisors function (A034448).

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A323341 Numbers k whose unitary divisors have an even sum which is larger than 2k, but they cannot be partitioned into two disjoint parts whose sums are equal.

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A335215 Bi-unitary Zumkeller numbers: numbers whose set of bi-unitary divisors can be partitioned into two disjoint sets of equal sum.

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A339979 Coreful Zumkeller numbers: numbers whose set of coreful divisors can be partitioned into two disjoint sets of equal sum.

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A290467 Unitary half-Zumkeller numbers: numbers k whose unitary proper divisors can be partitioned into two disjoint sets whose sums are equal.

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