A171641 -id:A171641 - cp's OEIS frontend

A337738 Terms of A171641 with a record number of divisors.

738, 3492, 14184, 58896, 236448, 954432, 2549700, 10884600, 44989200

Offset: 1

Amiram Eldar, Sep 17 2020

Non-deficient numbers (A023196) with an even sum of divisors (A000203) which cannot be partitioned into two disjoint sets with equal sum, and having a record number of divisors.

The corresponding numbers of divisors are 12, 18, 24, 30, 36, 42, 54, 72, 90, ...

			The number of divisors of each of the first 33 terms of A171641 is 12. A171641(34) = 3492 has 18 divisors, and it is the first term with more than 12 divisors. Therefore, a(2) = 3492.

Cf. A000005, A000203, A023196, A083207, A171641, A335008.

nonZumQ[n_] := Module[{d = Divisors[n], sum, x}, sum = Plus @@ d; sum >= 2*n && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] == 0]; dm = 0; s = {}; Do[d = DivisorSigma[0, n]; If[d > dm, q = nonZumQ[n]; If[q && d > dm, dm = d; AppendTo[s, n]]], {n, 1, 60000}]; s

a(8)-a(9) from Amiram Eldar, Apr 04 2023

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

A156903 Abundant numbers (A005101) whose abundance is odd.

Original entry on oeis.org

18, 36, 72, 100, 144, 162, 196, 200, 288, 324, 392, 400, 450, 576, 648, 784, 800, 882, 900, 968, 1152, 1296, 1352, 1458, 1568, 1600, 1764, 1800, 1936, 2178, 2304, 2450, 2500, 2592, 2704, 2916, 3042, 3136, 3200, 3528, 3600, 3872, 4050, 4356, 4608, 4624

Offset: 1

Robert G. Wilson v, Feb 17 2009

nonn

Equivalently, abundant numbers with odd sum of divisors.
Complement of A204825 with respect to A005101 (abundant numbers).
Seems to be a proper subset of A083211. - Robert G. Wilson v, Mar 30 2010. This sequence is indeed a proper subset of A083211, since the abundance of a number k, A033880(k) = sigma(k) - 2*k, has the same parity as sigma(k). If sigma(k) is odd then the sums of any two complementary subsets of the divisors of k have different parities and thus they cannot be equal. - Amiram Eldar, Jun 20 2020
If n is present, so is 2*n. - Robert G. Wilson v, Jun 21 2015
If n is in the sequence, so is 100*n (conjectured). - Sergey Pavlov, Mar 22 2017. Pavlov's observation trivially follows from the fact that to have odd abundance a number k must be either a square or twice a square. If such a number k is abundant then 100*k = (10^2) * k is abundant as well and has odd abundance. In general, we can say that if k is present, so are t^2*k and 2*t^2*k, for every t>0. - Giovanni Resta, Oct 16 2018
Terms are congruent to {0, 2, 4, 8, 9, 14, 16, 18, 20, 26, 28, 32} (mod 36). - Robert G. Wilson v, Dec 09 2018

			k = 18 is in the sequence because its divisors are {1,2,3,6,9,18} which sum to sigma(k) = 39; so its abundance is sigma(k) - 2k = 39 - 36 = 3.

Robert G. Wilson v, Table of n, a(n) for n = 1..22927 (corrected by Michel Marcus)
Eric Weisstein's World of Mathematics, Abundant Number

Intersection of A005101 and A028982. - Amiram Eldar, Jun 20 2020
Cf. A000203, A033880, A259231.
A proper subset of A083211. Setwise difference A083211 \ A171641.
Positions of odd negative terms in A378600.
Cf. A204825 (abundant numbers with even sum of divisors), A204826 (deficient numbers with odd sum of divisors), A204827 (deficient numbers with even sum of divisors).

Filtered([1..5000],k->Sigma(k)-2*k>0 and (Sigma(k)-2*k) mod 2=1); # Muniru A Asiru, Dec 11 2018

with(numtheory): select(k->sigma(k)>2*k and modp(sigma(k)-2*k,2)=1,[$1..5000]); # Muniru A Asiru, Dec 11 2018

abundance[n_] := DivisorSigma[1, n] - 2 n; Select[Range[1000], abundance[#] > 0 && Mod[abundance[#], 2] == 1 &]
abundOddAbundQ[n_] := If[MemberQ[{0, 2, 4, 8, 9, 14, 16, 18, 20, 26, 28, 32}, Mod[n, 36]], a = DivisorSigma[1, n]; OddQ@a && a > 2 n]; Select[ Range@ 5000, abundOddAbundQ@# &] (* Robert G. Wilson v, Dec 23 2018 *)

is(n)=my(k=sigma(n)-2*n); k>0 && k%2 \\ Charles R Greathouse IV, Feb 21 2017

from sympy.ntheory import divisor_sigma
def a(n):
    return divisor_sigma(n) - 2*n
[n for n in range(18, 5001) if a(n) > 0 and a(n) % 2] # Indranil Ghosh, Mar 22 2017

Name edited by Michel Marcus and Charles R Greathouse IV, Mar 26 2017

Edited by N. J. A. Sloane, Jun 21 2020 at the suggestion of Amiram Eldar

A083211 Abundant numbers (A005101) with no subset of their divisors such that the complement has the same sum.

Original entry on oeis.org

18, 36, 72, 100, 144, 162, 196, 200, 288, 324, 392, 400, 450, 576, 648, 738, 748, 774, 784, 800, 846, 882, 900, 954, 968, 1062, 1098, 1152, 1206, 1278, 1296, 1314, 1352, 1422, 1458, 1494, 1568, 1600, 1602, 1746, 1764, 1800, 1818, 1854, 1926, 1936, 1962, 2034, 2178, 2286, 2304, 2358, 2450, 2466, 2500, 2502, 2592, 2682, 2704, 2718, 2826, 2916, 2934, 3006, 3042

Offset: 1

Reinhard Zumkeller, Apr 22 2003

nonn

A083206(a(n)) = 0; subsequence of A083210.

All [abundant] numbers with an odd sum of divisors (either a square or twice a square, A028982) must be terms because for these numbers the two subsets will be of opposite parity. - Robert G. Wilson v, Apr 01 2010, clarified by Antti Karttunen, Dec 05 2024

			Divisors of n=18: {1,2,3,6,9,18}; 18 is pseudo-perfect (A005835): 18=9+6+3, but there exist no two complementary subsets of divisors having the same sum, therefore 18 is a term.

David A. Corneth, Table of n, a(n) for n = 1..12915 (first 6061 terms from Antti Karttunen)
Eric Weisstein's World of Mathematics, Abundant Number.
Reinhard Zumkeller, Illustration of initial terms

Intersection of A005101 and A083210.
Disjoint union of A156903 and A171641. - Amiram Eldar, Jun 20 2020
Positions of negative terms in A378600.
Cf. A000203, A028982, A083206, A156942 (odd terms), A378661 (characteristic function).

fQ[n_] := Block[{d = Divisors[n], t, ds, x}, ds = Total[d]; If[Mod[ds, 2] > 0, False, t = CoefficientList[Product[1 + x^i, {i, d}], x]; t[[1 + ds/2]] > 0]]; Select[Range[3042], And[DivisorSigma[1, #] > 2 #, ! fQ[#]] &] (* Michael De Vlieger, Dec 04 2024, after T. D. Noe at A083207 *)

A083206(n) = { my(s=sigma(n),p=1); if(s%2 || s < 2*n, 0, fordiv(n, d, p *= ('x^d + 'x^-d)); (polcoeff(p, 0)/2)); };
is_A083211(n) = ((sigma(n)>2*n) && (0==A083206(n))); \\ Antti Karttunen, Dec 04 2024

{k such that sigma(k) > 2*k and A083206(k) = 0}. - Antti Karttunen, Dec 04 2024

a(21)-a(46) from Robert G. Wilson v, Apr 01 2010

Many missing terms inserted, first ones at a(29) = 1206 and a(30) = 1278 - Antti Karttunen, Dec 04 2024

A378600 Signed variant of Zumkeller deficiency: a(n) = signum(A033879(n)) * A103977(n).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 5, 2, 10, 0, 12, 4, 6, 1, 16, -1, 18, 0, 10, 8, 22, 0, 19, 10, 14, 0, 28, 0, 30, 1, 18, 14, 22, -1, 36, 16, 22, 0, 40, 0, 42, 4, 12, 20, 46, 0, 41, 7, 30, 6, 52, 0, 38, 0, 34, 26, 58, 0, 60, 28, 22, 1, 46, 0, 66, 10, 42, 0, 70, -1, 72, 34, 26, 12, 58, 0, 78, 0, 41, 38, 82, 0, 62, 40, 54, 0, 88, 0, 70

Offset: 1

Antti Karttunen, Dec 04 2024

sign

If n is abundant, then negate the value of A103977(n), otherwise use as it is.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A033879, A103977.

Cf. A005100 (positions of terms > 0), A083207 (positions of 0's), A083211 (positions of negative terms), A156903 (positions of odd negative terms), A171641 (of even negative terms).

A033879(n) = (n+n-sigma(n));
nonzerocoefpositions(p) = { my(v=Vec(p), lista=List([])); for(i=1,#v,if(v[i], listput(lista,i))); Vec(lista); };
A103977(n) = { my(p=1); fordiv(n, d, p *= (1 + 'x^d)); my(plist=nonzerocoefpositions(p), m = #plist, d); if(!(m%2), plist[1+(m/2)]-plist[m/2], d = plist[(m+1)/2]-plist[(m-1)/2]; if(1==d,0,d)); };
A378600(n) = { my(d=A033879(n)); if(d>=0, d, -A103977(n)); };

If A033879(n) >= 0, a(n) = A033879(n), otherwise a(n) = -A103977(n).

A323342 Numbers k whose bi-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

704, 1458, 2394, 7544, 10184, 46400, 60416, 106434, 115182, 118098, 121014, 125000, 129762, 141426, 147258, 150174, 156006, 158922, 164754, 176418, 185166, 190998, 199746, 202662, 217242, 220158, 228906, 237654, 243486, 246402, 252234, 260982, 263898, 278478

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The bi-unitary version of A171641.

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

Cf. A171641, A188999, A292982, A323341, A323343, A323344.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[n_] := Select[Divisors[n], Last@Intersection[f@#, f[n/#]] == 1 &]; fun[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[n_] := If[n==1, 1, Times @@ (fun @@@ FactorInteger[n])]; seq={}; Do[s=bsigma[n]; If[OddQ[s] || s<=2n, Continue[]]; div = bdiv[n]; If[Coefficient[Times @@ (1 + x^div) // Expand, x, s/2] == 0, AppendTo[seq, n]], {n, 1, 10000}]; seq

A323343 Numbers k whose exponential 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

1910412, 9552060, 21014532, 24835356, 32477004, 43939476, 55401948, 59222772, 70685244, 78326892, 82147716, 89789364, 101251836, 105072660, 112714308, 116535132, 124176780, 127997604, 135639252, 139460076, 150922548, 158564196, 162385020, 170026668, 185309964

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The exponential version of A171641.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A051377, A129575, A171641, A323341, A323342, A323344.

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]]]} ]]; ediv[n_] := Module[ {d=Rest[Divisors[n]]}, Select[ d, expDivQ[n, #]&]]; esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[ Last[#]]}] &) /@ FactorInteger[n]; seq={}; Do[s=esigma[n]; If[OddQ[s] || s<=2n, Continue[]]; div = ediv[n]; If[Coefficient[Times @@ (1 + x^div) // Expand, x, s/2] == 0, AppendTo[seq, n]], {n, 1, 10000}]; seq

A323344 Numbers k whose infinitary 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, 7544, 10184, 1452330, 2154584, 5021912, 5747994, 5771934, 5786298, 5800662, 5834178, 5843754, 5858118, 5886846, 5905998, 5920362, 5929938, 5992182, 6035274, 6059214, 6078366, 6087942, 6102306, 6107094, 6121458, 6174126, 6202854, 6207642, 6245946, 6265098

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The infinitary version of A171641.

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

Cf. A049417, A077609, A129656, A171641, A323341, A323342, A323343.

infdivs[x_] := If[x == 1, 1, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[x] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] ; fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[n_] := If[n == 1, 1, Times @@ (fun @@@ FactorInteger[n])]; seq={}; Do[s=isigma[n]; If[OddQ[s] || s<=2n, Continue[]]; div = infdivs[n]; If[Coefficient[Times @@ (1 + x^div) // Expand, x, s/2] == 0, AppendTo[seq, n]], {n, 1, 100000}]; seq (* 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

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

cp's OEIS Frontend

A337738 Terms of A171641 with a record number of divisors.

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

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A156903 Abundant numbers (A005101) whose abundance is odd.

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A083211 Abundant numbers (A005101) with no subset of their divisors such that the complement has the same sum.

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A378600 Signed variant of Zumkeller deficiency: a(n) = signum(A033879(n)) * A103977(n).

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A323342 Numbers k whose bi-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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A323343 Numbers k whose exponential 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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