A119347 -id:A119347 - cp's OEIS frontend

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

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

A103977 Zumkeller deficiency of n: Let d_1 ... d_k be the divisors of n. Then a(n) = min_{ e_1 = +-1, ... e_k = +-1 } | Sum_i e_i d_i |.

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

Offset: 1

Yasutoshi Kohmoto, Jan 01 2007

nonn

Like the ordinary deficiency (A033879) obtains 0's only at perfect numbers (A000396), the Zumkeller deficiency obtains 0's only at integer-perfect numbers, A083207. See the formula section. Unlike the ordinary deficiency, this obtains only nonnegative values. See A378600 for another version. - Antti Karttunen, Dec 03 2024

			a(6) = 1 + 2 + 3 - 6 = 0.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 10000 terms from Amiram Eldar)

Cf. A125732, A125733, A005835, A023196, A033879, A083206, A083207 (positions of 0's), A263837, A378643 (Dirichlet inverse), A378644 (Möbius transform), A378645, A378646, A378647 (an analog of A000027), A378648 (an analog of sigma), A378649 (an analog of Euler phi), A379503 (positions of 1's), A379504, A379505.
Cf. A378600 (signed variant).
Cf. also A058377, A119347.

A103977 := proc(n) local divs,a,acandid,filt,i,p,sigs ; divs := convert(numtheory[divisors](n),list) ; a := add(i,i=divs) ; for sigs from 0 to 2^nops(divs)-1 do filt := convert(sigs,base,2) ; while nops(filt) < nops(divs) do filt := [op(filt), 0] ; od ; acandid := 0 ; for p from 0 to nops(divs)-1 do if op(p+1,filt) = 0 then acandid := acandid-op(p+1,divs) ; else acandid := acandid+op(p+1,divs) ; fi ; od: acandid := abs(acandid) ; if acandid < a then a := acandid ; fi ; od: RETURN(a) ; end: seq(A103977(n),n=1..80) ; # R. J. Mathar, Nov 27 2007
# second Maple program:
a:= proc(n) option remember; local l, b; l, b:= [numtheory[divisors](n)[]],
      proc(s, i) option remember; `if`(i<1, s,
        min(b(s+l[i], i-1), b(abs(s-l[i]), i-1)))
      end: b(0, nops(l))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Dec 05 2024

a[n_] := Module[{d = Divisors[n], c, p, m}, c = CoefficientList[Product[1 + x^i, {i, d}], x]; p = -1 + Position[c, ?(# > 0 &)] // Flatten; m = Length[p]; If[OddQ[m], If[(d = p[[(m + 1)/2]] - p[[(m - 1)/2]]) == 1, 0, d], p[[m/2 + 1]] - p[[m/2]]]]; Array[a, 100] (* _Amiram Eldar, Dec 11 2019 *)

nonzerocoefpositions(p) = { my(v=Vec(p), lista=List([])); for(i=1,#v,if(v[i], listput(lista,i))); Vec(lista); }; \\ Doesn't need to be 0-based, as we use their differences only.
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)); }; \\ Antti Karttunen, Dec 03 2024, after Mathematica-program by Amiram Eldar

If n=p (prime), then a(n)=p-1. If n=2^m, then a(n)=1. [Corrected by R. J. Mathar, Nov 27 2007]
a(n) = 0 iff n is a Zumkeller number (A083207). - Amiram Eldar, Jan 05 2020
From Antti Karttunen, Dec 03 2024: (Start)
a(n) = A033879(n) iff n is a non-abundant number (A263837).
a(n) = abs(A378600(n)).
a(n) = 2*A378647(n) - A378648(n). [Analogously to A033879(n) = 2*n - sigma(n)]
a(n) = 0 <=> A083206(n) > 0.
(End)
a(p^e) = p^e - (1+p+...+p^(e-1)) = (p^e*(p-2) + 1)/(p-1) for prime p. - Jianing Song, Dec 05 2024
a(n) = 1 <=> A379504(n) > 0. - Antti Karttunen, Jan 07 2025

More terms from R. J. Mathar, Nov 27 2007

Name "Zumkeller deficiency" coined by Antti Karttunen, Dec 03 2024

A100587 Number of nonempty subsets of divisors of n.

Original entry on oeis.org

1, 3, 3, 7, 3, 15, 3, 15, 7, 15, 3, 63, 3, 15, 15, 31, 3, 63, 3, 63, 15, 15, 3, 255, 7, 15, 15, 63, 3, 255, 3, 63, 15, 15, 15, 511, 3, 15, 15, 255, 3, 255, 3, 63, 63, 15, 3, 1023, 7, 63, 15, 63, 3, 255, 15, 255, 15, 15, 3, 4095, 3, 15, 63, 127, 15, 255, 3, 63, 15, 255, 3, 4095, 3

Offset: 1

Labos Elemer, Dec 01 2004

A119347(n) <= a(n). - Reinhard Zumkeller, Jun 27 2015

			For all prime numbers p, a(p)=3, since those subsets are {{1,p},{1},{p}}.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Wasin So, Integral circulant graphs, Discr. Math. 306 (1) (2006) 153-158

Cf. A000005, A066781, A100371.

Cf. A119347.

a100587 = (subtract 1) . (2 ^) . a000005'
-- Reinhard Zumkeller, Jun 27 2015

A100587:=n->-1+2^numtheory[tau](n): seq(A100587(n), n=1..100); # Wesley Ivan Hurt, Dec 12 2015

Table[2^DivisorSigma[0, n] - 1, {n, 73}] (* Michael De Vlieger, Dec 11 2015 *)

a(n) = 2^(numdiv(n)) - 1; \\ Michel Marcus, Dec 15 2013

a(n) = -1 + 2^tau(n), where tau(n) = DivisorSigma(0, n) = A000005(n).

A030057 Least number that is not a sum of distinct divisors of n.

Original entry on oeis.org

2, 4, 2, 8, 2, 13, 2, 16, 2, 4, 2, 29, 2, 4, 2, 32, 2, 40, 2, 43, 2, 4, 2, 61, 2, 4, 2, 57, 2, 73, 2, 64, 2, 4, 2, 92, 2, 4, 2, 91, 2, 97, 2, 8, 2, 4, 2, 125, 2, 4, 2, 8, 2, 121, 2, 121, 2, 4, 2, 169, 2, 4, 2, 128, 2, 145, 2, 8, 2, 4, 2, 196, 2, 4, 2, 8, 2, 169, 2, 187, 2, 4, 2, 225, 2, 4, 2, 181

Offset: 1

David W. Wilson

a(n) = 2 if and only if n is odd. a(2^n) = 2^(n+1). - Emeric Deutsch, Aug 07 2005
a(n) > n if and only if n belongs to A005153, and then a(n) = sigma(n) + 1. - Michel Marcus, Oct 18 2013
The most frequent values are 2 (50%), 4 (16.7%), 8 (5.7%), 13 (3.2%), 16 (2.4%), 29 (1.3%), 32 (1%), 40, 43, 61, ... - M. F. Hasler, Apr 06 2014
The indices of records occur at the highly abundant numbers, excluding 3 and 10, if Jaycob Coleman's conjecture at A002093 that all these numbers are practical numbers (A005153) is true. - Amiram Eldar, Jun 13 2020

			a(10)=4 because 4 is the least positive integer that is not a sum of distinct divisors (namely 1,2,5 and 10) of 10.

David Wasserman and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 1000 terms from David Wasserman)

Cf. A002093, A005153, A093896, A119347.
Distinct elements form A030058.
Cf. A027750.

a030057 n = head $ filter ((== 0) . p (a027750_row n)) [1..] where
   p _      0 = 1
   p []     _ = 0
   p (k:ks) x = if x < k then 0 else p ks (x - k) + p ks x
-- Reinhard Zumkeller, Feb 27 2012

with(combinat): with(numtheory): for n from 1 to 100 do div:=powerset(divisors(n)): b[n]:=sort({seq(sum(div[i][j],j=1..nops(div[i])),i=1..nops(div))}) od: for n from 1 to 100 do B[n]:={seq(k,k=0..1+sigma(n))} minus b[n] od: seq(B[n][1],n=1..100); # Emeric Deutsch, Aug 07 2005

a[n_] :=  First[ Complement[ Range[ DivisorSigma[1, n] + 1], Total /@ Subsets[ Divisors[n]]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 02 2012 *)

from sympy import divisors
def A030057(n):
    c = {0}
    for d in divisors(n,generator=True):
        c |=  {a+d for a in c}
    k = 1
    while k in c:
        k += 1
    return k # Chai Wah Wu, Jul 05 2023

Edited by N. J. A. Sloane, May 05 2007

A119348 Triangle read by rows: row n contains, in increasing order, all the distinct sums of distinct divisors of n.

Original entry on oeis.org

1, 1, 2, 3, 1, 3, 4, 1, 2, 3, 4, 5, 6, 7, 1, 5, 6, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 3, 4, 9, 10, 12, 13, 1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 1, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18

Offset: 1

Emeric Deutsch, May 15 2006

Row n contains A119347(n) terms. In row n the first term is 1 and the last term is sigma(n) (=sum of the divisors of n =A000203(n)). If row n contains all numbers from 1 to sigma(n), then n is called a practical number (A005153).

			Row 5 is 1,5,6, the possible sums obtained from the divisors 1 and 5 of 5.
Triangle starts:
1;
1,2,3;
1,3,4;
1,2,3,4,5,6,7;
1,5,6;
1,2,3,4,5,6,7,8,9,10,11,12;
1,7,8;
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15;
1,3,4,9,10,12,13;

T. D. Noe, Rows n=1..100, flattened

Cf. A000203, A005153, A119347.

with(numtheory): with(linalg): sums:=proc(n) local dl,t: dl:=convert(divisors(n),list): t:=tau(n): {seq(innerprod(dl,convert(2^t+i,base,2)[1..t]),i=1..2^t-1)} end: for n from 1 to 12 do sums(n) od; # yields sequence in triangular form

row[n_] := Union[Total /@ Subsets[Divisors[n]]] // Rest;
Table[row[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Aug 06 2024 *)

A237290 Sum of positive numbers k <= sigma(n) that are a sum of any subset of distinct divisors of n.

Original entry on oeis.org

1, 6, 8, 28, 12, 78, 16, 120, 52, 144, 24, 406, 28, 192, 192, 496, 36, 780, 40, 903, 256, 288, 48, 1830, 124, 336, 320, 1596, 60, 2628, 64, 2016, 384, 432, 384, 4186, 76, 480, 448, 4095, 84, 4656, 88, 2688, 2184, 576, 96, 7750, 228, 2976, 576, 3136, 108, 7260

Offset: 1

Jaroslav Krizek, Mar 02 2014

nonn

			For n = 5, a(5) = 1 + 5 + 6 = 12 (each of the numbers 1, 5 and 6 is the sum of a subset of distinct divisors of 5).
The numbers n = 14 and 15 is an interesting pair of consecutive numbers with identical value of sigma(n) such that simultaneously a(14) = a(15) and A237289(14) = A237289(15).
a(14) = 1+2+3+7+8+9+10+14+15+16+17+21+22+23+24 = a(15) = 1+3+4+5+6+8+9+15+16+18+19+20+21+23+24 = 192.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 200 terms from Vincenzo Librandi)
Jon Maiga, Computer-generated formulas for A237290, Sequence Machine.

Cf. A000203, A119348, A005153, A119347 (count of the same numbers), A184387, A229335, A237287, A237289.

isSumDist := proc(n,k)
    local dvs,s ;
    dvs := numtheory[divisors](n) ;
    for s in combinat[powerset](dvs) do
        add(m,m=op(s)) ;
        if % = k then
            return true;
        end if;
    end do:
    false ;
end proc:
A237290 := proc(n)
    local a;
    a := 0 ;
    for k from 1 to numtheory[sigma](n) do
        if isSumDist(n,k) then
            a := a+k;
        end if;
    end do:
end proc:
seq(A237290(n),n=1..20) ; # R. J. Mathar, Mar 13 2014

a[n_] := Plus @@ Union[Plus @@@ Subsets@ Divisors@ n]; Array[a, 54] (* Giovanni Resta, Mar 13 2014 *)

padbin(n, len) = {b = binary(n); while(length(b) < len, b = concat(0, b);); b;}
a(n) = {vks = []; d = divisors(n); nbd = #d; for (i=1, 2^nbd-1, b = padbin(i, nbd); onek = sum(j=1, nbd, d[j]*b[j]); vks = Set(concat(vks, onek));); sum(i=1, #vks, vks[i]);} \\ Michel Marcus, Mar 09 2014

A237290(n) = { my(c=[0]); fordiv(n,d, c = Set(concat(c,vector(#c,i,c[i]+d)))); vecsum(c); }; \\ after Chai Wah Wu's Python-code, Antti Karttunen, Nov 29 2024

from sympy import divisors
def A237290(n):
    ds = divisors(n)
    c, s = {0}, sum(ds)
    for d in ds:
        c |=  {a+d for a in c}
    return sum(a for a in c if 1<=a<=s) # Chai Wah Wu, Jul 05 2023

a(n) = A184387(n) - A237289(n).
a(p) = 2(p+2) for odd primes p.
a(n) = A184387(n) for practical numbers n (A005153), a(n) < A184387(n) for numbers n that are not practical (A237287).
a(n) = A000203(n) * (A119347(n)+1) / 2. [Found by Sequence Machine and easily seen to be true. Compare for example to the formulas of A229335.] - Antti Karttunen, Nov 29 2024

A193279 Number of distinct sums of distinct proper divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 3, 7, 1, 16, 1, 7, 7, 15, 1, 21, 1, 22, 7, 7, 1, 36, 3, 7, 7, 28, 1, 42, 1, 31, 7, 7, 7, 55, 1, 7, 7, 50, 1, 54, 1, 31, 27, 7, 1, 76, 3, 31, 7, 31, 1, 66, 7, 64, 7, 7, 1, 108, 1, 7, 29, 63, 7, 78, 1, 31, 7, 72, 1, 123, 1, 7, 31, 31

Offset: 1

Michael Engling, Jul 20 2011

nonn

a(n)=1 if and only if n is prime.
a(n)=n-1 if n is a power of 2.
a(n)=n if n is an even perfect number (is the converse true?)
Note: the count excludes an empty subset of proper divisors that would give 0 as a sum. - Antti Karttunen, Mar 07 2018

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 10000 terms from Amiram Eldar)

Cf. A193280.

Cf. A119347 (allows also n to be included in the sums), A378447 (differences).

with(linalg): a:=proc(n) local dl,t: dl:=convert(numtheory[divisors](n) minus {n}, list): t:=nops(dl): return nops({seq(innerprod(dl, convert(2^t+i, base, 2)[1..t]), i=1..2^t-1)}): end: seq(a(n), n=1..76); # Nathaniel Johnston, Jul 23 2011

a[n_] := Module[{d = Most @ Divisors[n], x}, Count[CoefficientList[Product[1 + x^i, {i, d}], x], ?(# > 0 &)] - 1]; Array[a, 100] (* _Amiram Eldar, Jun 13 2020 *)

\\ Slow and naive:
A193279(n) = if(1==n,0,my(pds = (divisors(n)[1..(numdiv(n)-1)]), maxsum = vecsum(pds), sums = vector(maxsum), psetsiz = (2^length(pds))-1, k = 0, s); for(i=1,psetsiz,s = vecsum(choosebybits(pds,i)); if(!sums[s],k++;sums[s]++)); (k)); \\ Antti Karttunen, Mar 07 2018

A193279(n) = { my(p=1); fordiv(n, d, if(dAntti Karttunen, Nov 29 2024

A193279(n) = { my(c=[0]); fordiv(n,d, if(dA119347) - Antti Karttunen, Nov 29 2024

cp's OEIS Frontend

A378602 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A119347(i) = A119347(j), for all i, j >= 1.

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A378451 Dirichlet inverse of A119347, where A119347(n) is the number of distinct sums of distinct divisors of n.

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A378596 Möbius transform of A119347, where A119347 number of distinct sums of nonempty subsets of divisors of n.

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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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A103977 Zumkeller deficiency of n: Let d_1 ... d_k be the divisors of n. Then a(n) = min_{ e_1 = +-1, ... e_k = +-1 } | Sum_i e_i d_i |.

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A100587 Number of nonempty subsets of divisors of n.

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A030057 Least number that is not a sum of distinct divisors of n.

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