A111592 -id:A111592 - cp's OEIS frontend

A329190 Weird admirable numbers: numbers that are both weird (A006037) and admirable (A111592).

70, 836, 4030, 5830, 7192, 7912, 10792, 17272, 45356, 83312, 91388, 113072, 243892, 254012, 388076, 786208, 1713592, 4145216, 4199030, 4632896, 9928792, 11547352, 13086016, 15126992, 17999992, 29465852, 29581424, 34869056, 74899952, 89283592, 95327216, 120888092

Offset: 1

Amiram Eldar, Nov 07 2019

nonn

Admirable numbers that are not pseudoperfect (A005835).

Differs from A258250 at n >= 13.

Amiram Eldar, Table of n, a(n) for n = 1..66 (terms below 10^11)

Intersection of A006037 and A111592.

Cf. A005835, A258250.

admQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2]; weirdQ[n_] := Module[{d = Most[Divisors[n]]}, If[Total[d] <= n, False, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 0]]; Select[Range[10000], admQ[#] && weirdQ[#] &]

A364727 Numbers k such that k and k+1 are both admirable numbers (A111592).

Original entry on oeis.org

29691198404, 478012798575, 2789405835075, 22929723392715

Offset: 1

Amiram Eldar, Aug 05 2023

a(1)-a(2) were found by Giovanni Resta.

Giovanni Resta, admirable numbers, Numbers Aplenty.

Subsequence of A096399 and A111592.

Cf. A109729, A109730.

a(4) from Martin Ehrenstein, Aug 06 2023

A329187 Initial members of admirable triples: numbers k such that (k, k+2, k+4) are all admirable numbers (A111592).

Original entry on oeis.org

364, 17272, 54064, 383404, 1820812, 1945804, 2466604, 3283024, 3503164, 3820684, 3907984, 4407952, 5553712, 6095344, 6320524, 6820492, 7845232, 7966252, 8591212, 9841132, 10990864, 11841004, 12428272, 13695052, 13903372, 15032272, 15569932, 19007212, 19740304

Offset: 1

Amiram Eldar, Nov 07 2019

nonn

			364 is in the sequence since 364, 366, and 368 are all admirable numbers.

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

Subsequence of A109730, A111592, and A231088.

admQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2]; k = 0; s = {}; Do[If[admQ[n], k++; If[k > 2, AppendTo[s, n - 4]], k = 0], {n, 6, 2*10^6, 2}]; s (* for even terms; after Shyam Sunder Gupta at A231088 *)

A109138 Numbers n such that 2^n + 2 is an admirable number (A111592).

Original entry on oeis.org

6, 8, 12, 14, 18, 20, 24, 32, 44, 62, 80, 102, 128, 168, 192, 200, 314, 348, 702

Offset: 1

Jason Earls, Aug 18 2005

For k > 1, A000978(k)+1 is a member. Are there any others? - David Wasserman, May 28 2008

No more terms below 1064. - Amiram Eldar, Oct 12 2019

			a(3)=12 because 2^12 + 2 = 4098 and 1+2+3+683+1366+2049-6 = 4098.

Cf. A000978.

fQ[n_] := Block[{d = Most[ Divisors[n]], k = 1}, l = Length[d]; s = Plus @@ d; While[k < l && s - 2d[[k]] > n, k++ ]; If[k > l || s != n + 2d[[k]], False, True]]; Do[ If[ fQ[2^n + 2], Print[n]], {n, 200}] (* Robert G. Wilson v, Aug 30 2005 *)

More terms from David Wasserman, May 28 2008

a(19) from Amiram Eldar, Oct 12 2019

A364975 Admirable numbers (A111592) with a record gap to the next admirable number.

Original entry on oeis.org

12, 30, 42, 88, 120, 140, 186, 534, 678, 6774, 7962, 77118, 94108, 152826, 478194, 662154, 935564, 1128174, 2028198, 6934398, 7750146, 8330924, 9984738, 10030804, 22956114, 62062566, 151040622, 284791602, 732988732, 804394974, 1151476732, 9040886574, 31302713634

Offset: 1

Amiram Eldar, Aug 15 2023

nonn

The corresponding record gaps are 8, 10, 12, 14, 18, 34, 36, 48, 84, 132, 204, 216, 254, 312, 348, 360, 392, 468, 516, 528, 552, 598, 624, 638, 828, 852, 936, 1056, 1082, 1128, 1454, 1692, 1752, ... .

			The first 5 admirable numbers are 12, 20, 24, 30 and 40. The differences between these terms are 8, 4, 6 and 10. The record gaps, 8 and 10, occur after the terms 12 and 30, which are the first two terms of this sequence.

Cf. A111592, A364727.

Similar sequences: A306953, A330870, A334418, A334419, A334883, A363296.

admQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2];
seq[kmax_] := Module[{s = {}, m = 12, dm = 0}, Do[If[admQ[k], d = k - m; If[d > dm, dm = d; AppendTo[s, m]]; m = k], {k, m + 1, kmax}]; s]; seq[10^6]

isadm(n) = {my(ab=sigma(n)-2*n); ab>0 && ab%2 == 0 && ab/2 < n && n%(ab/2) == 0; }
lista(kmax) = {my(m = 12, dm = 0); for(k = m+1, kmax, if(isadm(k), d = k - m; if(d > dm, dm = d; print1(m, ", ")); m = k));}

A083207 Zumkeller or integer-perfect numbers: numbers n whose divisors can be partitioned into two disjoint sets with equal sum.

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, 228, 234, 240, 246, 252, 258, 260, 264, 270, 272

Offset: 1

Reinhard Zumkeller, Apr 22 2003

The 229026 Zumkeller numbers less than 10^6 have a maximum difference of 12. This leads to the conjecture that any 12 consecutive numbers include at least one Zumkeller number. There are 1989 odd Zumkeller numbers less than 10^6; they are exactly the odd abundant numbers that have even abundance, A174865. - T. D. Noe, Mar 31 2010
For k >= 0, numbers of the form 18k + 6 and 18k + 12 are terms (see Remark 2.3. in Somu et al., 2023). Corollary: The maximum difference between any two consecutive terms is at most 12. - Ivan N. Ianakiev, Jan 02 2024
All 205283 odd abundant numbers less than 10^8 that have even abundance (see A174865) are Zumkeller numbers. - T. D. Noe, Nov 14 2010
Except for 1 and 2, all primorials (A002110) are Zumkeller numbers (follows from Fact 6 in the Rao/Peng paper). - Ivan N. Ianakiev, Mar 23 2016
Supersequence of A111592 (follows from Fact 3 in the Rao/Peng paper). - Ivan N. Ianakiev, Mar 20 2017
Conjecture: Any 4 consecutive terms include at least one number k such that sigma(k)/2 is also a Zumkeller number (verified for the first 10^5 Zumkeller numbers). - Ivan N. Ianakiev, Apr 03 2017
LeVan studied these numbers using the equivalent definition of numbers n such that n = Sum_{d|n, dA180332) "minimal integer-perfect numbers". - Amiram Eldar, Dec 20 2018
The numbers 3 * 2^k for k > 0 are all Zumkeller numbers: half of one such partition is {3*2^k, 3*2^(k-2), ...}, replacing 3 with 2 if it appears. With this and the lemma that the product of a Zumkeller number and a number coprime to it is again a Zumkeller number (see A179527), we have that all numbers divisible by 6 but not 9 (or numbers congruent to 6 or 12 modulo 18) are Zumkeller numbers, proving that the difference between consecutive Zumkeller numbers is at most 12. - Charlie Neder, Jan 15 2019
Improvements on the previous comment: 1) For every integer q > 0, every odd integer r > 0 and every integer s > 0 relatively prime to 6, the integer 2^q*3^r*s is a Zumkeller number, and therefore 2) there exist Zumkeller numbers divisible by 9 (such as 54, 90, 108, 126, etc.). - Ivan N. Ianakiev, Jan 16 2020
Conjecture: If d > 1, d|k and tau(d)*sigma(d) = k, then k is a Zumkeller number (cf. A331668). - Ivan N. Ianakiev, Apr 24 2020
This sequence contains A378541, the intersection of the practical numbers (A005153) with numbers with even sum of divisors (A028983). - David A. Corneth, Nov 03 2024
Sequence gives the positions of even terms in A119347, and correspondingly, of odd terms in A308605. - Antti Karttunen, Nov 29 2024
If s = sigma(m) is odd and p > s then m*p is not in the sequence. - David A. Corneth, Dec 07 2024

			Given n = 48, we can partition the divisors thus: 1 + 3 + 4 + 6 + 8 + 16 + 24 = 2 + 12 + 48, therefore 48 is a term (A083206(48) = 5).
From _David A. Corneth_, Dec 04 2024: (Start)
30 is in the sequence. sigma(30) = 72. So we look for distinct divisors of 30 that sum to 72/2 = 36. That set or its complement contains 30. The other divisors in that set containing 30 sum to 36 - 30 = 6. So we look for some distinct proper divisors of 30 that sum to 6. That is from the divisors of {1, 2, 3, 5, 6, 10, 15}. It turns out that both 1+2+3 and 6 satisfy this condition. So 36 is in the sequence.
25 is not in the sequence as sigma(25) = 31 which is odd so the sum of two equal integers cannot be the sum of divisors of 25.
33 is not in the sequence as sigma(33) = 48 < 2*33. So is impossible to have a partition of the set of divisors into two disjoint set the sum of each of them sums to 48/2 = 24 as one of them contains 33 > 24 and any other divisors are nonnegative. (End)

Marijo O. LeVan, Integer-perfect numbers, Journal of Natural Sciences and Mathematics, Vol. 27, No. 2 (1987), pp. 33-50.
Marijo O. LeVan, On the order of nu(n), Journal of Natural Sciences and Mathematics, Vol. 28, No. 1 (1988), pp. 165-173.
J. Sandor and B. Crstici, Handbook of Number Theory, II, Springer Verlag, 2004, chapter 1.10, pp. 53-54.

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Basher, k-Zumkeller labeling of super subdivision of some graphs, J. Egyptian Math. Soc. (2021) Vol. 29, No. 12.
Hussein Behzadipour, Two-layered numbers, arXiv:1812.07233 [math.NT], 2018.
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, arXiv:0912.0052 [math.NT], 2009.
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.
David A. Corneth, PARI programs (Use function "is" for A179527)
Bhabesh Das, On unitary Zumkeller numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 2, 436-442.
Farid Jokar, On the difference between Zumkeller numbers, arXiv:1902.02168 [math.NT], 2019.
Farid Jokar, On k-layered numbers and some labeling related to k-layered numbers, arXiv:2003.11309 [math.NT], 2020.
Farid Jokar, On k-layered numbers, arXiv:2207.09053 [math.NT], 2022.
Peter Luschny, Zumkeller Numbers.
Pankaj Jyoti Mahanta, Manjil P. Saikia, and Daniel Yaqubi, Some properties of Zumkeller numbers and k-layered numbers, arXiv:2008.11096 [math.NT], 2020.
Pankaj Jyoti Mahanta, Manjil P. Saikia, and Daniel Yaqubi, Some properties of Zumkeller numbers and k-layered numbers, Journal of Number Theory (2020).
Sai Teja Somu, Andrzej Kukla, and Duc Van Khanh Tran, Some Results on Zumkeller Numbers, arXiv:2310.14149 [math.NT], 2023.
Reinhard Zumkeller, Illustration of initial terms
Index entries for sequences where any odd perfect numbers must occur

Positions of nonzero terms in A083206, positions of 0's in A103977 and in A378600.
Positions of even terms in A119347, of odd terms in A308605.
Complement of A083210.
Subsequence of A023196 and of A028983.
Union of A353061 and A378541.
Subsequences: A000396, A083209, A118372, A180332, A293453, A334410, A378652, A378599.
Conjectured subsequences: A007691, A331668 (after their initial 1's), A351548 (apart from 0-terms).
Cf. A174865 (Odd abundant numbers whose abundance is even).
Cf. A204830, A204831 (equal sums of 3 or 4 disjoint subsets).
Cf. A000203, A005101, A005153 (practical numbers), A005835, A027750, A048055, A083206, A083208, A083211, A171641, A175592, A179527 (characteristic function), A221054.

a083207 n = a083207_list !! (n-1)
a083207_list = filter (z 0 0 . a027750_row) $ [1..] 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

with(numtheory): with(combstruct):
is_A083207 := proc(n) local S, R, Found, Comb, a, s; s := sigma(n);
if not(modp(s, 2) = 0 and n * 2 <= s) then return false fi;
S := s / 2 - n; R := select(m -> m <= S, divisors(n)); Found := false;
Comb := iterstructs(Combination(R)):
while not finished(Comb) and not Found do
   Found := add(a, a = nextstruct(Comb)) = S
od; Found end:
A083207_list := upto -> select(is_A083207, [$1..upto]):
A083207_list(272); # Peter Luschny, Dec 14 2009, updated Aug 15 2014

ZumkellerQ[n_] := Module[{d=Divisors[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[1000], ZumkellerQ] (* T. D. Noe, Mar 31 2010 *)
znQ[n_]:=Length[Select[{#,Complement[Divisors[n],#]}&/@Most[Rest[ Subsets[ Divisors[ n]]]],Total[#[[1]]]==Total[#[[2]]]&]]>0; Select[Range[300],znQ] (* Harvey P. Dale, Dec 26 2022 *)

part(n,v)=if(n<1, return(n==0)); forstep(i=#v,2,-1,if(part(n-v[i],v[1..i-1]), return(1))); n==v[1]
is(n)=my(d=divisors(n),s=sum(i=1,#d,d[i])); s%2==0 && part(s/2-n,d[1..#d-1]) \\ Charles R Greathouse IV, Mar 09 2014

PARI
```
\\ See Corneth link
```

from sympy import divisors
from sympy.combinatorics.subsets import Subset
for n in range(1,10**3):
    d = divisors(n)
    s = sum(d)
    if not s % 2 and max(d) <= s/2:
        for x in range(1,2**len(d)):
            if sum(Subset.unrank_binary(x,d).subset) == s/2:
                print(n,end=', ')
                break
# Chai Wah Wu, Aug 13 2014

from sympy import divisors
import numpy as np
A083207 = []
for n in range(2,10**3):
    d = divisors(n)
    s = sum(d)
    if not s % 2 and 2*n <= s:
        d.remove(n)
        s2, ld = int(s/2-n), 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:
                A083207.append(n)
                break
# Chai Wah Wu, Aug 19 2014

def is_Zumkeller(n):
    s = sigma(n)
    if not (2.divides(s) and n*2 <= s): return False
    S = s // 2 - n
    R = (m for m in divisors(n) if m <= S)
    return any(sum(c) == S for c in Combinations(R))
A083207_list = lambda lim: [n for n in (1..lim) if is_Zumkeller(n)]
print(A083207_list(272)) # Peter Luschny, Sep 03 2018

A083206(a(n)) > 0.
A083208(n) = A083206(a(n)).
A179529(a(n)) = 1. - Reinhard Zumkeller, Jul 19 2010

Name improved by T. D. Noe, Mar 31 2010

Name "Zumkeller numbers" added by N. J. A. Sloane, Jul 08 2010

A109729 Odd admirable numbers: such that sigma(n) = 2n + 2d for some d | n.

Original entry on oeis.org

945, 4095, 6435, 7425, 8415, 8925, 9555, 26145, 28035, 30555, 31815, 32445, 43065, 46035, 78975, 80535, 81081, 103455, 129195, 182655, 191565, 261261, 279279, 351351, 354585, 355725, 371925, 403095, 411255, 430815, 437745, 442365, 458055

Offset: 1

Jason Earls, Aug 09 2005

nonn

Equivalently: Odd n such that sigma(n)/2 - n is a positive divisor of n. (Negative and/or half-integer d = sigma(n)/2 - n, of which n could be a multiple, are excluded. Negative d correspond to deficient n, half-integer d to square n: the first example of an abundant n being a multiple of a half-integer d is n = 13167^2 = 173369889.) - M. F. Hasler, Jan 26 2020

Amiram Eldar, Table of n, a(n) for n = 1..746 (terms below 3*10^12)

Cf. A111592 (admirable numbers).

Cf. A000203 (sigma: sum of divisors).

Select[Range[1,460000,2],MemberQ[2#+2 Divisors[#],DivisorSigma[1,#]]&] (* Harvey P. Dale, Mar 23 2025 *)

select( {is_A109729(n,s=sigma(n))=s>2*n&&n%(s/2-n)==0&&n%2&&!(s%2)}, [2*k-1|k<-[1..5e5\2]]) \\ M. F. Hasler, Jan 26 2020

Offset corrected by Amiram Eldar, Jun 22 2019

Name edited by M. F. Hasler, Jan 26 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]

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));

A334972 Bi-unitary admirable numbers: numbers k such that there is a proper bi-unitary divisor d of k such that bsigma(k) - 2d = 2k, where bsigma is the sum of bi-unitary divisors function (A188999).

Original entry on oeis.org

24, 30, 40, 42, 48, 54, 56, 66, 70, 78, 80, 88, 102, 104, 114, 120, 138, 150, 162, 174, 186, 222, 224, 246, 258, 270, 282, 294, 318, 354, 360, 366, 402, 420, 426, 438, 448, 474, 498, 534, 540, 582, 606, 618, 630, 642, 654, 660, 672, 678, 720, 726, 762, 780, 786

Offset: 1

Amiram Eldar, May 18 2020

nonn

Equivalently, numbers that are equal to the sum of their proper bi-unitary divisors, with one of them taken with a minus sign.

Admirable numbers (A111592) that are exponentially odd (A268335) are also bi-unitary admirable numbers since all of their divisors are bi-unitary. Terms that are not exponentially odd are 48, 80, 150, 162, 294, 360, 420, 448, 540, 630, 660, 720, 726, 780, 832, 990, ...

			48 is in the sequence since 48 = 1 + 2 + 3 - 6 + 8 + 16 + 24 is the sum of its proper bi-unitary divisors with one of them, 6, taken with a minus sign.

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

The bi-unitary version of A111592.
Subsequence of A292982.
Cf. A188999, A222266, A268335, A328328, A334974.

fun[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), (p^(e + 1) - 1)/(p - 1) - p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); buDivQ[n_, 1] = True; buDivQ[n_, div_] := If[Mod[#2, #1] == 0, Last@Apply[Intersection, Map[Select[Divisors[#], Function[d, CoprimeQ[d, #/d]]] &, {#1, #2/#1}]] == 1, False] & @@ {div, n}; buAdmQ[n_] := (ab = bsigma[n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && buDivQ[n, ab/2]; Select[Range[1000], buAdmQ]

cp's OEIS Frontend

A329190 Weird admirable numbers: numbers that are both weird (A006037) and admirable (A111592).

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A364727 Numbers k such that k and k+1 are both admirable numbers (A111592).

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A329187 Initial members of admirable triples: numbers k such that (k, k+2, k+4) are all admirable numbers (A111592).

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A109138 Numbers n such that 2^n + 2 is an admirable number (A111592).

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A364975 Admirable numbers (A111592) with a record gap to the next admirable number.

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PARI

A083207 Zumkeller or integer-perfect numbers: numbers n whose divisors can be partitioned into two disjoint sets with equal sum.

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A109729 Odd admirable numbers: such that sigma(n) = 2n + 2d for some d | n.

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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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A334972 Bi-unitary admirable numbers: numbers k such that there is a proper bi-unitary divisor d of k such that bsigma(k) - 2d = 2k, where bsigma is the sum of bi-unitary divisors function (A188999).