Zumkeller Numbers and
Zumkeller Partitions ≤ 4000

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, 228,
234, 240, 246, 252, 258, 260, 264, 270, 272, 276, 280, 282, 294, 300, 304,
306, 308, 312, 318, 320, 330, 336, 340, 342, 348, 350, 352, 354, 360, 364,
366, 368, 372, 378, 380, 384, 390, 396, 402, 408, 414, 416, 420, 426, 432,
438, 440, 444, 448, 456, 460, 462, 464, 468, 474, 476, 480, 486, 490, 492,
496, 498, 500, 504, 510, 516, 520, 522, 528, 532, 534, 540, 544, 546, 550,
552, 558, 560, 564, 570, 572, 580, 582, 588, 594, 600, 606, 608, 612, 616,
618, 620, 624, 630, 636, 640, 642, 644, 650, 654, 660, 666, 672, 678, 680,
684, 690, 696, 700, 702, 704, 708, 714, 720, 726, 728, 732, 736, 740, 744,
750, 756, 760, 762, 768, 770, 780, 786, 792, 798, 804, 810, 812, 816, 820,
822, 828, 832, 834, 836, 840, 852, 858, 860, 864, 868, 870, 876, 880, 888,
894, 896, 906, 910, 912, 918, 920, 924, 928, 930, 936, 940, 942, 945, 948,
952, 960, 966, 972, 978, 980, 984, 990, 992, 996, 1000

A positive integer n is said to be a Zumkeller number [A083207] if the positive factors of n can be partitioned into two disjoint parts so that the sums of the two parts are equal. We shall call such a partition a Zumkeller partition. (K.P.S. Bhaskara Rao, Yuejian Peng, On Zumkeller Numbers.)

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Semi-Zumkeller Numbers ≤ 20000

738, 748, 774, 846, 954, 1062, 1098, 1206, 1278, 1314, 1422,1494, 1602,
1746, 1818, 1854, 1926, 1962, 2034, 2286, 2358, 2466, 2502, 2682, 2718,
2826, 2934, 3006, 3114, 3222, 3258, 3438, 3474, 3492, 3546, 3582, 3636,
3708, 3798, 3852, 3924, 4014, 4068, 4086, 4122, 4194, 4302, 4338, 4518,
4572, 4626, 4716, 4734, 4842, 4878, 4932, 4986, 5004, 5058, 5094, 5274,
5364, 5436, 5526, 5598, 5634, 5652, 5706, 5868, 5958, 6012, 6066, 6228,
6246, 6282, 6354, 6444, 6462, 6516, 6606, 6714, 6822, 6876, 6894, 6948,
7002, 7092, 7146, 7164, 7218, 7362, 7542, 7544, 7578, 7596, 7758, 7794,
7902, 7974, 8028, 8082, 8172, 8226, 8244, 8298, 8334, 8388, 8406, 8604,
8622, 8676, 8766, 8838, 8982, 9036, 9054, 9162, 9252, 9378, 9414, 9468,
9684, 9738, 9756, 9846, 9972

Semi-Zumkeller numbers are non-deficient numbers (numbers n such that sigma(n) ≥ 2n) [A023196] such that sigma(n) is even [A152678], but which are not Zumkeller numbers [A083207 ], i.e., the positive factors of n cannot be partitioned into two disjoint parts so that the sums of the two parts are equal.

Conjecture 1: Semi-Zumkeller numbers are even numbers.
Conjecture 2: Semi-Zumkeller numbers are semi-perfect numbers [A005835].
Conjecture 3: Semi-Zumkeller which are not divisible by 3 are strange. The only strange Semi-Zumkeller numbers below 20000 are 748 , 7544, 10184.

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A special type of Zumkeller numbers.

Numbers n such that sigma(n) = 2(n + sum(prime divisors of n)).

532, 945, 2624, 5704, 6536, 229648, 497696, 652970, 685088, 997408

The definition amounts to the existence of a partition
of the divisors of n, as follows.

Let
[A000203]   sigma (n) = sum{1<=d<=n, d|n},
[A105221]   sigma'(n) = sum{1< d <n, d|n, d prime},
[A060278]   sigma*(n) = sum{1< d <n, d|n, d not prime}.
Then
   sigma(n) = 1 + sigma'(n) + sigma*(n) + n.

A member of the sequence cannot be a prime. Thus
sum(prime_divisors(n)) has to be sigma'(n).
Therefore n is a term of the sequence if and only if
   1 + sigma*(n) = sigma'(n) + n.
This shows that n is a Zumkeller number.

Example: 532 is a member of the sequence because
  1  +  sigma*(532) = 1+4+14+28+38+76+133+266 = 560,
  sigma'(532) + 532  = 2+7+19+532 = 560.

Maple.
   is_a := proc(x) local k;
   sigma(x) = 2*(x+add(k,k=select(isprime,divisors(x)))) end;
   select(is_a,[$1..3000]);

A variant of the Zumkeller condition:
The sum of the even divisors is twice the sum of the odd divisors.

(A171642) Numbers which are non-deficient (2n <= sigma(n)) [A023196] such that sigma(n) [A000203] is odd and the sum of the even divisors [A074400] is twice the sum of the odd divisors [A000593].
The sequence of terms a(n) which have not the form 72n^2+72n+18 starts: 2450, 6050, 8450, 61250, 120050, 151250, 211250, 296450.
Example: divisors(18) = {1, 2, 3, 6, 9, 18},
sigma(18) = 39, and 2 + 6 + 18 = 2*(1 + 3 + 9).

18, 162, 450, 882, 1458, 2178, 2450, 3042, 4050, 5202,
6050, 6498, 7938, 8450, 9522, 11250, 13122, 15138, 17298,
19602, 22050, 24642, 27378, 30258, 33282, 36450, 39762,
43218, 46818, 50562, 54450, 58482, 61250, 62658, 66978,
71442, 76050, 80802, 85698, 90738, 95922, 101250, 106722,
112338, 118098, 120050, 124002, 130050, 136242, 142578,
149058, 151250, 155682, 162450, 169362, 176418, 183618,
190962, 198450, 206082, 211250, 213858, 221778, 229842,
238050, 246402, 254898, 263538, 272322, 281250, 290322,
296450, 299538

with(numtheory): A171642 := proc(n) local k,s,a;
s := sigma(n); a := add(k,k=select(x->type(x,odd),divisors(n)));
if 3*a = s and 2*n <= s and type(s,odd) then n else NULL fi end:

An observation of Frank Buss

Frank Buss plotted the difference of consecutive Zumkeller numbers up to 27188. An interesting picture emerged. The greatest distance between two consecutive Zumkeller numbers in this region is 12.

Image by Frank Buss (Creative Commons License).

Implementations

Maple

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