A083207 -id:A083207 - cp's OEIS frontend

A204830 Numbers k whose divisors can be partitioned into three disjoint sets whose sums are all sigma(k)/3.

120, 180, 240, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 960, 1080, 1260, 1320, 1440, 1512, 1560, 1584, 1620, 1680, 1800, 1848, 1890, 1920, 1980, 2016, 2040, 2160, 2184, 2280, 2340, 2352, 2376, 2400, 2520, 2640, 2688, 2760, 2772, 2856, 2880, 2940, 3000

Offset: 1

Jaroslav Krizek, Jan 22 2012

nonn

Subsequence of the intersection of A023197 and A087943.
If m is a term then so is m*p^k when p is coprime to m. - David A. Corneth, Mar 09 2024
Is this sequence equal to the sequence: "Numbers k such that sigma(k) is divisible by 3 and sigma(k) >= 3*k"? - David A. Corneth, Mar 17 2024
Answer: No. The numbers k with sigma(k) >= 3k and sigma(k) divisible by 3 that are not in this sequence are in A306476. - Amiram Eldar, Jun 22 2024

			180 is a term because sigma(180)/3 = 182 = 2 + 180 = 1+3+4+5+6+9+10+15+18+30+36+45 = 12+20+60+90 (summands are all the divisors of 180).

David A. Corneth, Table of n, a(n) for n = 1..10296 (first 500 terms from Paolo P. Lava, terms <= 550000).
Farid Jokar, On k-layered numbers, arXiv:2207.09053 [math.NT], 2022.

Cf. A023197, A083207 (Zumkeller numbers -- numbers k whose divisors can be partitioned into two disjoint sets whose sums are both sigma(k)/2), A087943, A204831 (numbers k whose divisors can be partitioned into four disjoint sets whose sums are all sigma(k)/4), A306476.

A211225 Number of ways to represent sigma(n) as sigma(x) + sigma(y) where x+y = n.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 1, 2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2

Offset: 1

Paolo P. Lava, May 07 2012

nonn

From an idea of Charles R Greathouse IV.

a(A211223(n)) > 0. - Reinhard Zumkeller, Jan 06 2013

			a(3)=1 because sigma(3)=sigma(1)+sigma(2)=4;
a(32)=2 because sigma(32)=sigma(4)+sigma(28)=sigma(14)+sigma(18)=63;
a(117)=3 because sigma(117)=sigma(41)+sigma(76)=sigma(52)+sigma(65)=sigma(56)+sigma(61)=182; etc.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A083207, A204830, A204831, A211223, A211224.

Cf. A000203.

a211225 n = length $ filter (== a000203 n) $ zipWith (+) us' vs where
   (us,vs@(v:_)) = splitAt (fromInteger $ (n - 1) `div` 2) a000203_list
   us' = if even n then v : reverse us else reverse us
-- Reinhard Zumkeller, Jan 06 2013

with(numtheory);
A211225:=proc(q)
local b,i,n;
for n from 1 to q do
  b:=0;
  for i from 1 to trunc(n/2) do
    if sigma(i)+sigma(n-i)=sigma(n) then b:=b+1; fi;
  od;
  print(b)
od; end:
A211225(1000);

a[n_] := With[{s = DivisorSigma[1, n]}, Sum[Boole[s == DivisorSigma[1, x] + DivisorSigma[1, n-x]], {x, 1, Quotient[n, 2]}]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 04 2023 *)

a(n)=my(t=sigma(n)); sum(i=1, n\2, sigma(i)+sigma(n-i)==t) \\ Charles R Greathouse IV, May 07 2012

A083210 Numbers with no subset of their divisors such that the complement has the same sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91

Offset: 1

Reinhard Zumkeller, Apr 22 2003

nonn

A083206(a(n)) = 0; complement of A083207; deficient numbers (A005100) are a subset.

A179529(a(n)) = 0. [Reinhard Zumkeller, Jul 19 2010]

Antti Karttunen, Table of n, a(n) for n = 1..20000
Reinhard Zumkeller, Illustration of initial terms

Positions of 0's in A083206.
Cf. A083207 (complement).
Cf. A005100, A083211 (subsequences).

is_A083210(n) = !A083206(n); \\ Antti Karttunen, Dec 02 2024

from itertools import count, islice
from sympy import divisors
def A083210_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        d = divisors(n)
        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:
                    break
            else:
                yield n
        else:
            yield n
A083210_list = list(islice(A083210_gen(),30)) # Chai Wah Wu, Feb 13 2023

A204831 Numbers n whose divisors can be partitioned into four disjoint sets whose sums are all sigma(n)/4.

Original entry on oeis.org

27720, 30240, 32760, 50400, 55440, 60480, 65520, 75600, 83160, 85680, 90720, 95760, 98280, 100800, 105840, 110880, 115920, 120120, 120960, 128520, 131040, 138600, 143640, 151200, 163800, 166320, 171360, 180180, 181440, 184800, 191520

Offset: 1

Jaroslav Krizek, Jan 22 2012

nonn

Subsequence of A023198 (numbers n such that sigma(n) >= 4n).

			Number 27720 is in the sequence because sigma(27720)/4 = 28080 = 360 + 27720 = 20 + 60 + 280 + 2310 + 4620 + 6930 + 13860 = 9 + 30 + 420 + 1540 + 1980 + 2772 + 3080 + 3465 + 5544 + 9240 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 10 + 11 + 12 + 14 + 15 + 18 + 21 + 22 + 24 + 28 + 33 + 35 + 36 + 40 + 42 + 44 + 45 + 55 + 56 + 63 + 66 + 70+ 72 + 77 + 84 + 88 + 90 + 99 + 105 + 110 + 120 + 126 + 132 + 140 + 154 + 165 + 168 + 180 + 198 + 210 + 220 + 231 + 252 + 264+ 308 + 315 + 330 + 385 + 396 + 440 + 462 + 495 + 504 + 616 + 630 + 660 + 693 + 770 + 792 + 840 + 924 + 990 + 1155 + 1260 + 1320 + 1386 + 1848 + 2520 + 3960 (summands are all divisors of 27720).

Farid Jokar, On k-layered numbers, arXiv:2207.09053 [math.NT], 2022.

Cf. A083207 (Zumkeller numbers--numbers n whose divisors can be partitioned into two disjoint sets whose sums are both sigma(n)/2), A204830 (numbers n whose divisors can be partitioned into three disjoint sets whose sums are all sigma(n)/3).

with(numtheory);with(combstruct);
A204831:=proc(i)
local S,R,Stop,Comb,c,d,k,m,n,s;
for n from 1 to i do
  s:=sigma(n); c:=op(divisors(n));
  if (modp(s,4)=0 and 4*n<=s) then
     S:=1/4*s-n; R:=select(m->m<=S,[c]); Stop:=false;
     Comb:=iterstructs(Combination(R));
     while not (finished(Comb) or Stop) do
       Stop:=add(d,d=nextstruct(Comb))=S;
     od;
     if Stop then print(n); fi;
  fi;
od;
end:
A204831(100000); # Paolo P. Lava, Jan 24 2012

A211224 Least k with precisely n partitions k = x + y satisfying sigma(k) = sigma(x) + sigma(y).

Original entry on oeis.org

3, 32, 117, 183, 393, 728, 933, 2193, 2528, 1173, 6136, 2990, 4070, 8211, 11488, 12616, 6112, 22287, 20584, 37468, 38675, 35245, 41416, 55825, 43616, 66385, 56810, 94040, 88736, 93975, 90068, 174515, 169376, 146965, 139196, 210453, 140576, 177248

Offset: 1

Paolo P. Lava, May 04 2012

nonn

Subset of A211223.

			a(7)=933; 933 has 7 partitions of two numbers, x and y, for which sigma(933) = sigma(x) + sigma(y) = 1248. In fact:
sigma(311) + sigma(622) = 312 + 936 = 1248;
sigma(322) + sigma(611) = 576 + 672 = 1248;
sigma(370) + sigma(563) = 684 + 564 = 1248;
sigma(391) + sigma(542) = 432 + 816 = 1248;
sigma(398) + sigma(535) = 600 + 648 = 1248;
sigma(407) + sigma(526) = 456 + 792 = 1248;
sigma(442) + sigma(491) = 756 + 492 = 1248;
Furthermore 933 is the minimum number to have this property.

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A083207, A204830, A204831, A211223, A211225.

with(numtheory);
A211224:=proc(q)
local a,b,i,j,n,v;
v:=array(1..10000); for n from 1 to 10000 do v[n]:=0; od;
a:=0;
for n from 1 to q do
  b:=0;
  for i from 1 to trunc(n/2) do
    if sigma(i)+sigma(n-i)=sigma(n) then b:=b+1; fi; od;
  if b=a+1 then a:=b; print(n); j:=1;
     while v[b+j]>0 do a:=b+j; print(v[b+j]); j:=j+1; od;
  else if b>a+1 then if v[b]=0 then v[b]:=n; fi; fi; fi;
od; end:
A211224(1000);

ct(n)=my(t=sigma(n));sum(i=1,n\2,sigma(i)+sigma(n-i)==t)
v=vector(100);for(n=1,1e5,t=ct(n);if(t&&t<=#v&&!v[t],v[t]=n));v
\\ Charles R Greathouse IV, May 04 2012

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

A349753 Odd numbers k for which A003961(k)-2k divides A003961(k)-sigma(k), where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 7, 25, 33, 55, 57, 69, 91, 93, 2211, 4825, 12639, 28225, 32043, 68727, 89575, 970225, 2245557, 16322559, 22799825, 48980427, 55037217, 60406599, 68258725, 325422273, 414690595, 569173299, 794579511, 10056372275, 10475647197, 10759889913, 11154517557

Offset: 1

Antti Karttunen, Dec 01 2021

nonn

Numbers k for which A326057(k) = gcd(A003961(k)-2k, A003961(k)-sigma(k)) is equal to abs(A252748(k)) = |A003961(k)-2k|.
The odd terms of A326134 form a subsequence of this sequence. Unlike in A326134, here we don't constrain the value of A252748(k) = A003961(k)-2k, thus allowing also values <= +1. Because of that, the odd terms of A048674 and A348514 are all included here, for example 57 and 68727 that occur in A348514, and 1, 3, 25, 33, 93, 970225, 325422273, 414690595 that occur in A048674.
Conjecture (1): This is a subsequence of A319630, in other words, for all terms k, gcd(k, A003961(k)) = 1.
Conjecture (2): Apart from 1, there are no common terms with A349169, which would imply that no odd perfect numbers exist.
None of the 36 initial terms is Zumkeller, in A083207, because all are deficient (in A005100). See also A337372. - Antti Karttunen, Dec 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..36 (terms below 10^11)
Index entries for sequences where any odd perfect numbers must occur
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A003961, A005100, A083207, A252748, A286385, A319630, A326057, A326134, A337372, A349169.
Cf. A048674, A348514.
Subsequence of A378980 (its odd terms).

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := Divisible[(sn = s[n]) - DivisorSigma[1, n], sn - 2*n]; Select[Range[1, 10^6, 2], q] (* Amiram Eldar, Dec 04 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349753(n) = if(!(n%2), 0, my(s = A003961(n), t = (s-(2*n)), u = s-sigma(n)); !(u%t));

A379503 Almost Zumkeller numbers: Numbers whose Zumkeller-deficiency (A103977) is 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 18, 32, 36, 64, 72, 100, 128, 144, 162, 196, 200, 256, 288, 324, 392, 400, 450, 512, 576, 648, 784, 800, 882, 900, 968, 1024, 1152, 1296, 1352, 1458, 1568, 1600, 1764, 1800, 1936, 2048, 2178, 2304, 2450, 2500, 2592, 2704, 2916, 3042, 3136, 3200, 3528, 3600, 3872, 4050, 4096, 4356, 4608, 4624, 4900, 5000

Offset: 1

Antti Karttunen, Jan 06 2025

nonn

Numbers whose divisors can be partitioned into two disjoint sets with equal sum when an extra 1-divisor is added to them. - Amiram Eldar, Jan 06 2025

Question: Does A156942 give all odd squares > 1 of this sequence? There are two issues here: first, whether there are any almost perfect numbers (k such that sigma(k) = 2k-1) that are odd (and by necessity squares) other than 1, and second, whether A103977(k) = 1 for all terms of A156942? The first 15000 terms of A156942 are all members.

			18 is included, as its divisors with an extra 1 are [1, 1, 2, 3, 6, 9, 18], and these can be partitioned as 2+3+6+9 = 1+1+18 = 20.
36 is included, as its divisors with an extra 1 are [1, 1, 2, 3, 4, 6, 9, 12, 18, 36], and these can be partitioned to two sets with equal sums, for example as (1+2+3+4)+(36) = (1+9)+(6+12+18), and also in several other ways (see example in A379504).
11025 is included as its divisors with an extra 1 are [1, 1, 3, 5, 7, 9, 15, 21, 25, 35, 45, 49, 63, 75, 105, 147, 175, 225, 245, 315, 441, 525, 735, 1225, 1575, 2205, 3675, 11025], and 1+5+35+175+245+11025 = 1+3+7+9+15+21+25+45+49+63+75+105+147+225+315+441+525+735+1225+1575+2205+3675 = 11486 = (sigma(11025)+1)/2.

Robert Israel, Table of n, a(n) for n = 1..2500 (n = 1..430 from Antti Karttunen)

Positions of 1's in A103977, positions of nonzero terms in A379504.
Cf. A083207, A379502 (characteristic function).
Subsequences: A000079, A156942 (conjectured)
Subsequence of A028982, and of A083210.

KK:= proc(S) # Karmarkar-Karp algorithm
  local R,n,a,b;
  R:= S;
  for n from nops(R) by -1 to 2 do
    R:= sort([abs(R[-1]-R[-2]), op(R[1..-3])]);
  od;
  op(R) = 0
end proc:
filter:= proc(n) local S,t,d,R,i;
  S:= [1, op(numtheory:-divisors(n))];
  t:= convert(S,`+`)/2;
  if t < n then return false fi;
  if not t::integer then return false fi;
  if KK(S) then return true fi;
  evalb(coeff(mul(1+x^d,d=S),x,t) <> 0)
end proc;
select(filter, [$1..10000]); # Robert Israel, Jan 06 2025

PARI
```
is_A379503 = A379502;
```

A011774 Nonprimes k that divide sigma(k) + phi(k).

Original entry on oeis.org

1, 312, 560, 588, 1400, 23760, 59400, 85632, 147492, 153720, 556160, 569328, 1590816, 2013216, 3343776, 4563000, 4695456, 9745728, 12558912, 22013952, 23336172, 30002960, 45326160, 52021242, 75007400, 113315400, 137617728, 153587720, 402831360, 699117024

Offset: 1

R. K. Guy

2*k = sigma(k) + phi(k) if and only if k is 1 or a prime.
If 7*2^j - 1 is prime then m = 2^(j+2)*3*(7*2^j - 1) is in the sequence. Because phi(m) = 2^(j+2)*(7*2^j - 2); sigma(m) = 7*2^(j+2)*(2^(j+3) - 1) so phi(m) + sigma(m) = 2^(j+2)*((7*2^j - 2) + (7*2^(j+3) - 7)) = 2^(j+2)* (63*2^(j+2) - 9) = 3*(2^(j+2)*3*(7*2^j - 1)) = 3*m, hence m is a term of A011251 and consequently m is a term of this sequence. A112729 gives such m's. - Farideh Firoozbakht, Dec 01 2005
Conjecture: For n > 1, a(n) is a Zumkeller number (A083207). Verified for all n in [2,63]. - Ivan N. Ianakiev, Jan 25 2023

			a(26) = 113315400: sigma = 426535200, phi = 26726400, quotient = 4.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B42, p. 151.
Zhang Ming-Zhi, typescript submitted to Unsolved Problems section of Monthly, 96-01-10.

Giovanni Resta, Table of n, a(n) for n = 1..63 (terms < 10^13; first 53 terms from Donovan Johnson)
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.
Q.-X. Jin and M. Tang, The 4-Nicol Numbers Having Five Different Prime Divisors, J. Int. Seq. 14 (2011) # 11.7.2.
Eric Weisstein's World of Mathematics, Prime Number.

Cf. A065387, A011251, A011254, A055681, A001771, A112729.

Do[If[Mod[DivisorSigma[1, n]+EulerPhi[n], n]==0, Print[n]], {n, 1, 2*10^7}]
Do[ If[ ! PrimeQ[n] && Mod[ DivisorSigma[1, n] + EulerPhi[n], n] == 0, Print[n] ], {n, 1, 10^8} ]

sp(n)=my(f=factor(n));n*prod(i=1, #f[,1], 1-1/f[i,1]) + prod(i=1, #f[,1], (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1))
p=2;forprime(q=3, 1e6, for(n=p+1, q-1, if(sp(n)%n==0, print1(n", ")));p=q) \\ Charles R Greathouse IV, Mar 19 2012

More terms from David W. Wilson

Corrected by Labos Elemer, Feb 12 2004

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).

cp's OEIS Frontend

A204830 Numbers k whose divisors can be partitioned into three disjoint sets whose sums are all sigma(k)/3.

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A211225 Number of ways to represent sigma(n) as sigma(x) + sigma(y) where x+y = n.

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A083210 Numbers with no subset of their divisors such that the complement has the same sum.

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A204831 Numbers n whose divisors can be partitioned into four disjoint sets whose sums are all sigma(n)/4.

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A211224 Least k with precisely n partitions k = x + y satisfying sigma(k) = sigma(x) + sigma(y).

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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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A349753 Odd numbers k for which A003961(k)-2k divides A003961(k)-sigma(k), where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

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A379503 Almost Zumkeller numbers: Numbers whose Zumkeller-deficiency (A103977) is 1.

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