A083207 -id:A083207 - cp's OEIS frontend

A221054 Numbers whose distinct prime factors can be partitioned into two equal sums.

1, 30, 60, 70, 90, 120, 140, 150, 180, 240, 270, 280, 286, 300, 350, 360, 450, 480, 490, 540, 560, 572, 600, 646, 700, 720, 750, 810, 900, 960, 980, 1080, 1120, 1144, 1200, 1292, 1350, 1400, 1440, 1500, 1620, 1750, 1798, 1800, 1920, 1960, 2145, 2160, 2240, 2250, 2288, 2310, 2400, 2430, 2450, 2584, 2700, 2730, 2800, 2880, 3000, 3135

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 15 2013

nonn

This is a superset of 2*product of twin primes, A071142.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Table of n, a(n), and equal sums of factors for n=1..10000

Cf. A175592 (multiplicity of prime factors allowed).
Cf. A071139-A071147, especially A071140.
Cf. A008472, A037074, A071142.
Cf. A027748, A005843, A083207.

a221054 n = a221054_list !! (n-1)
a221054_list = filter (z 0 0 . a027748_row) $ tail a005843_list where
   z u v []     = u == v
   z u v (p:ps) = z (u + p) v ps || z u (v + p) ps
-- Reinhard Zumkeller, Apr 18 2013

q[n_] := Module[{p = FactorInteger[n][[;; , 1]], sum, x}, sum = Total[p]; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, p}], x][[1 + sum/2]] > 0]; Select[Range[3200], q] (* Amiram Eldar, May 31 2025 *)

isok(k) = my(f=factor(k), nb=#f~); for (i=0,2^nb-1, my(v=Vec(Vecrev(binary(i)), nb)); if (sum(k=1, nb, if (v[k], f[k,1])) == sum(k=1, nb, if (!v[k], f[k,1])), return(1));); \\ Michel Marcus, May 31 2025

Missing terms inserted by Michel Marcus, May 31 2025

A335142 Nonunitary Zumkeller numbers: numbers whose set of nonunitary divisors is nonempty and can be partitioned into two disjoint sets of equal sum.

Original entry on oeis.org

24, 48, 54, 80, 96, 112, 120, 150, 160, 168, 180, 192, 216, 224, 240, 252, 264, 270, 280, 294, 312, 320, 336, 352, 360, 378, 384, 396, 408, 416, 432, 448, 456, 468, 480, 486, 504, 528, 540, 552, 560, 594, 600, 612, 624, 630, 640, 672, 684, 696, 702, 704, 720, 726

Offset: 1

Amiram Eldar, May 25 2020

nonn

Apparently, most of the terms are nonunitary abundant (A064597). Term that are nonunitary deficient (A064598) are 54, 150, 270, 294, 378, ...

			24 is a term since its set of nonunitary divisors, {2, 4, 6, 12}, can be partitioned into the two disjoint sets, {2, 4, 6} and {12}, whose sum is equal: 2 + 4 + 6 = 12.

Cf. A064591, A064597, A064598, A083207.

nuzQ[n_] := Module[{d = Select[Divisors[n], GCD[#, n/#] > 1 &], sum, x}, sum = Plus @@ d; sum > 0 && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; Select[Range[1000], nuzQ]

A335197 Infinitary Zumkeller numbers: numbers whose set of infinitary divisors can be partitioned into two disjoint sets of equal sum.

Original entry on oeis.org

6, 24, 30, 40, 42, 54, 56, 60, 66, 70, 72, 78, 88, 90, 96, 102, 104, 114, 120, 138, 150, 168, 174, 186, 210, 216, 222, 246, 258, 264, 270, 280, 282, 294, 312, 318, 330, 354, 360, 366, 378, 384, 390, 402, 408, 420, 426, 438, 440, 456, 462, 474, 480, 486, 498, 504

Offset: 1

Amiram Eldar, May 26 2020

nonn

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

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

The infinitary version of A083207.
Subsequence of A129656.
Cf. A077609, A323344.

infdivs[n_] := If[n == 1, {1}, Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; infZumQ[n_] := Module[{d = infdivs[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[500], infZumQ] (* after Michael De Vlieger at A077609 *)

A376880 Numbers that have Zumkeller divisors.

Original entry on oeis.org

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264, 270

Offset: 1

Peter Luschny, Oct 20 2024

nonn

d is a Zumkeller divisor of n if and only if d is a divisor of n and is Zumkeller (A083207).

The first difference from A023196 is 748, which is abundant (sigma(748) = 1512 > 2*748) but has no Zumkeller divisors.

			The Zumkeller divisors of 80 are {20, 40, 80}, so 80 is a term.
The divisors of 81 are {1, 3, 9, 27, 81}, none of which is Zumkeller, so 81 is not a term.

Peter Luschny, Table of n, a(n) for n = 1..10000

Cf. A083207, A023196, A171641, A376879, A376881.

Positions of terms > 1 in A376882, terms > 0 in A378446.

with(NumberTheory):
isZumkeller := proc(n) option remember; local s, p, i, P;
    s := SumOfDivisors(n);
    if s::odd or s < n*2 then false else
    P := mul(1 + x^i, i in Divisors(n));
    is(0 < coeff(P, x, s/2)) fi end:
select(n -> ormap(isZumkeller, Divisors(n)), [seq(1..270)]);

znQ[n_]:=Length[Select[{#, Complement[Divisors[n], #]}&/@Most[Rest[ Subsets[ Divisors[ n]]]], Total[#[[1]]]==Total[#[[2]]]&]]>0; zn=Select[Range[300], znQ] (* zn from A083207 *) ;Select[Range[270],IntersectingQ[Divisors[#],zn]&] (* James C. McMahon, Oct 23 2024 *)

Incorrect comment removed by Peter Luschny, Dec 02 2024

A246198 Half-Zumkeller numbers: numbers n whose proper positive divisors can be partitioned into two disjoint sets whose sums are equal.

Original entry on oeis.org

6, 12, 20, 24, 28, 30, 40, 42, 48, 54, 56, 60, 66, 70, 78, 80, 84, 88, 90, 96, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 150, 156, 160, 168, 174, 176, 180, 186, 192, 198, 204, 208, 210, 216, 220, 222, 224, 225, 228, 234, 240, 246, 252, 258, 260, 264

Offset: 1

Chai Wah Wu, Aug 18 2014

nonn

All even half-Zumkeller numbers are in A083207, i.e. they are Zumkeller numbers (see Clark et al. 2008). The first 47 terms coincide with A083207. 225 is the first number in the sequence that is not a Zumkeller number.

			Proper divisors of 225 are 1, 3, 5, 9, 15, 25, 45, 75 and 1+3+15+25+45=5+9+75.

S. Clark et al., Zumkeller numbers, Mathematical Abundance Conference, April 2008.

Robert Israel, Table of n, a(n) for n = 1..9188 (n=1..1309 from Chai Wah Wu)
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.
Pankaj Jyoti Mahanta, Manjil P. Saikia, and Daniel Yaqubi, Some properties of Zumkeller numbers and k-layered numbers, arXiv:2008.11096 [math.NT], 2020.

Cf. A083207.

filter:= proc(n) local L,s,t,nL,B,j,k;
   L:= numtheory:-divisors(n) minus {n};
   s:= convert(L,`+`);
   if s::odd then return false fi;
   t:= s/2;
   nL:= nops(L);
   B:= Array(0..t,1..nL);
   B[0,1]:= 1;
   B[L[1],1]:= 1;
   for j from 2 to nL do
      B[..,j]:= B[..,j-1];
      for k from L[j] to t do
         B[k,j]:= B[k,j] + B[k-L[j],j-1]
      od:
      if B[t,j] > 0 then return true fi;
   od:
   false
end:
select(filter, [$2..300]); # Robert Israel, Aug 19 2014

filterQ[n_] := Module[{L, s, t, nL, B, j, k},
  L = Most[Divisors[n]];
  s = Total[L];
  If[OddQ[s], Return[False]];
  t = s/2;
  nL = Length[L];
  B[, ] = 0;
  B[0, 1] = 1;
  B[L[[1]], 1] = 1;
  For[j = 2, j <= nL, j++,
    Do[B[k, j] = B[k, j-1], {k, 0, t}];
    For[k = L[[j]], k <= t, k++,
      B[k, j] = B[k, j] + B[k-L[[j]], j-1]
    ];
    If[ B[t, j] > 0, Return[True]];
  ];
  False
];
Select[Range[2, 300], filterQ] (* Jean-François Alcover, Mar 04 2019, after Robert Israel *)
hzQ[n_] := Module[{d = Most @ Divisors[n], sum, x}, sum = Plus @@ d; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; Select[Range[2, 1000], hzQ] (* Amiram Eldar, May 03 2020 *)

from sympy.combinatorics.subsets import Subset
from sympy import divisors
A246198 = []
for n in range(2,10**3):
    d = divisors(n)
    d.remove(n)
    s, dmax = sum(d), max(d)
    if not s % 2 and 2*dmax <= s:
        d.remove(dmax)
        s2 = s/2-dmax
        for x in range(2**len(d)):
            if sum(Subset.unrank_binary(x,d).subset) == s2:
                A246198.append(n)
                break

from sympy import divisors
import numpy as np
A246198 = []
for n in range(2, 10**3):
    d = divisors(n)
    d.remove(n)
    s, dmax = sum(d), max(d)
    if not s % 2 and 2*dmax <= s:
        d.remove(dmax)
        s2, ld = int(s/2-dmax), len(d)
        z = np.zeros((ld+1, s2+1), dtype=int)
        for i in range(1, ld+1):
            y = min(d[i-1], s2+1)
            z[i, range(y)] = z[i-1, range(y)]
            z[i, range(y, s2+1)] = np.maximum(z[i-1, range(y, s2+1)], z[i-1, range(0, s2+1-y)]+y)
            if z[i, s2] == s2:
                A246198.append(n)
                break
# Chai Wah Wu, Aug 19 2014

A292544 Numbers h such that 2^phi(h) == phi(h) (mod h).

Original entry on oeis.org

1, 12, 40, 48, 60, 192, 544, 640, 680, 704, 768, 816, 960, 1020, 1664, 3072, 10240, 11008, 12288, 13760, 15360, 19456, 24320, 49152, 83968, 125952, 131584, 139264, 139808, 163840, 164480, 174080, 174760, 196608, 197376, 208896, 209712, 245760, 246720, 261120, 262140, 720896, 786432

Offset: 1

Max Alekseyev and Altug Alkan, Sep 18 2017

Conjecture: For n > 1, a(n) is a Zumkeller number (A083207) [confirmed for n up to 47]. - Ivan N. Ianakiev, Sep 22 2017

			704 = 11*2^6 is a term since phi(11*2^6) = 5*2^6 and 11*2^6 divides 2^(5*2^6) - 5*2^6.

Giovanni Resta, Table of n, a(n) for n = 1..180 (terms < 10^12; first 101 terms from Michel Marcus)

Cf. A000010, A007733, A066781.

{1}~Join~Select[Range[10^6], Function[n, # == PowerMod[2, #, n] &@ EulerPhi@ n]] (* Michael De Vlieger, Sep 18 2017 *)

isok(n) = Mod(2, n)^eulerphi(n)==eulerphi(n);

Let m be an odd number, z = A007733(m) and k, 0 <= k < z, be such that phi(m) == 2^k (mod m); then m*2^(i*z - k + 1) belongs to this sequence for all i >= 1. And this is a general form of the terms of this sequence.
Some families of solutions of the form m*2^(i*z - k + 1):
If m = 3, then z = 2 and k = 1 ==> 3*2^(2*i) is a term for all i >= 1.
If m = 5, then z = 4 and k = 2 ==> 5*2^(4*i-1) is a term for all i >= 1.
If m = 7, then z = 3 but k does not exist ==> no term with odd part equal to 7.
If m = 15, then z = 4 and k = 3 ==> 15*2^(4*i-2) is a term for all i >= 1.
If m = 77, then z = 30 and k = 14 ==> 77*2^(30*i-13) is a term for all i >= 1.

A326133 Numbers n for which sigma(n) > A005187(n).

Original entry on oeis.org

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 110, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264, 270, 272, 276, 280, 282, 288, 294, 300

Offset: 1

Antti Karttunen, Jun 11 2019

nonn

Differs from A023196 for the first time at the 28th term, which here is 110, which is not included in A023196.

Note that as there is at least one odd number (815634435) in A326138, it means that A005231 does not contain all odd terms of this sequence.

Positions of negative terms in A294898.
Cf. A000203, A005187, A023196.
Cf. A000396, A005231, A083207, A111592, A326131, A326138 (subsequences).

Select[Range[300], DivisorSigma[1, #] > 2*# - DigitCount[2*#, 2, 1] &] (* Amiram Eldar, Aug 06 2023 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
isA326133(n) = (sigma(n)>A005187(n));

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]

A335218 Exponential Zumkeller numbers: numbers whose exponential divisors can be partitioned into two disjoint subsets of equal sum.

Original entry on oeis.org

36, 180, 252, 396, 468, 612, 684, 828, 900, 1044, 1116, 1260, 1332, 1476, 1548, 1692, 1764, 1800, 1908, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2988, 3060, 3204, 3276, 3420, 3492, 3600, 3636, 3708, 3852, 3924, 4068, 4140, 4284, 4356, 4500, 4572, 4716, 4788, 4900

Offset: 1

Amiram Eldar, May 27 2020

nonn

First differs from A318100 at n = 49: 4900 is a term that is not an exponential pseudoperfect number.

			36 is a term since its exponential divisors, {6, 12, 18, 36}, can be partitioned into 2 disjoint sets whose sum is equal: 6 + 12 + 18 = 36.

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

The exponential version of A083207.
Subsequence of A129575.
A054979 is a subsequence.
Cf. A318100, A322791, A323343.

dQ[n_, m_] := (n > 0 && m > 0 && Divisible[n, m]); expDivQ[n_, d_] := Module[{ft = FactorInteger[n]}, And @@ MapThread[dQ, {ft[[;; , 2]], IntegerExponent[d, ft[[;; , 1]]]}]]; eDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expDivQ[n, #] &]]; ezQ[n_] := Module[{d = eDivs[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[10^4], ezQ]

A347063 Double Zumkeller numbers: numbers whose set of divisors can be partitioned into two disjoint subsets with equal sums and equal cardinalities.

Original entry on oeis.org

24, 30, 42, 48, 54, 60, 66, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 140, 150, 156, 160, 168, 174, 180, 186, 192, 198, 204, 210, 216, 220, 222, 224, 228, 240, 246, 252, 258, 260, 264, 270, 276, 280, 282, 300, 306, 308, 312, 318, 320, 330, 336, 340, 342

Offset: 1

Ivan N. Ianakiev, Aug 15 2021

nonn

If x is a Zumkeller number, then so is 2x. Conjecturally, if y is a term of this sequence, then so is 2y.
If y is a term of this sequence, then so is p*y, where p is a prime that is coprime to y. Proof: Let D = {d_1,d_2,...,d_k} be the set of divisors of y. Let E be the set of divisors of p*y. Except for the divisors of y E contains their products with p. In other words, E = {d_1,d_2,...,d_2k}, meaning that the cardinality of E is twice the cardinality of D. Those additional divisors are F = {p*d_1,p*d_2,...,p*d_k}. Since D can be partitioned into two disjoint subsets with equal sums and cardinalities by definition, this must be true about F and also about E = D union F. QED. - Ivan N. Ianakiev, Nov 20 2021
It seems that for k>=1 all numbers of the form 18k+12 are terms. Verified for k in [1, 45]. - Ivan N. Ianakiev, Oct 01 2024

			The set of divisors of 24 is D = {1,2,3,4,6,8,12,24}. D = {1,2,3,24} union {4,6,8,12}, so 24 is in the sequence.

Subsequence of A083207 (Zumkeller numbers).

Select[Range@300,!IntegerQ@Sqrt@#&&(d=Divisors@#; MemberQ[Total/@Subsets[d,{Length@d/2}],Total@d/2])&] (* Giorgos Kalogeropoulos, Aug 15 2021 *)

More terms from Jinyuan Wang, Aug 15 2021

cp's OEIS Frontend

A221054 Numbers whose distinct prime factors can be partitioned into two equal sums.

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A335142 Nonunitary Zumkeller numbers: numbers whose set of nonunitary divisors is nonempty and can be partitioned into two disjoint sets of equal sum.

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A335197 Infinitary Zumkeller numbers: numbers whose set of infinitary divisors can be partitioned into two disjoint sets of equal sum.

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A376880 Numbers that have Zumkeller divisors.

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A246198 Half-Zumkeller numbers: numbers n whose proper positive divisors can be partitioned into two disjoint sets whose sums are equal.

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A292544 Numbers h such that 2^phi(h) == phi(h) (mod h).

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A326133 Numbers n for which sigma(n) > A005187(n).

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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.