cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

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Author

Reinhard Zumkeller, Apr 22 2003

Keywords

Comments

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

Examples

			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)

References

  • 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.

Links

Crossrefs

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.

Programs

  • Haskell
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Python
    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
  • Python
    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
  • Sage
    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

Formula

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

Extensions

Name improved by T. D. Noe, Mar 31 2010
Name "Zumkeller numbers" added by N. J. A. Sloane, Jul 08 2010

A382469 Numbers k such that the largest prime factor of k equals the sum of its remaining distinct prime factors.

Original entry on oeis.org

30, 60, 70, 90, 120, 140, 150, 180, 240, 270, 280, 286, 300, 350, 360, 450, 480, 490, 540, 560, 572, 600, 646, 700, 720, 750, 810, 900, 960, 980, 1080, 1120, 1144, 1200, 1292, 1350, 1400, 1440, 1500, 1620, 1750, 1798, 1800, 1920, 1960, 2160, 2240, 2250, 2288, 2400, 2430, 2450, 2584, 2700, 2800, 2880, 3000, 3135, 3146, 3240
Offset: 1

Views

Author

Paolo Xausa, Mar 31 2025

Keywords

Comments

A larger than usual number of terms is provided in order to distinguish this sequence from A365795, from which it first differs at n = 58 (a(58) = 3135 is missing from A365795).
First differs from A071140 at n = 140.

Links

Crossrefs

Positions of zeros in A382468.
Supersequence of A365795.
Cf. A006530, A027748, A071140, A221054.

Programs

  • Mathematica
    A382469Q[k_] := Last[#] == Total[Most[#]] & [FactorInteger[k][[All, 1]]];
Select[Range[4000], A382469Q]

A175592 Numbers n whose prime factors can be partitioned into two disjoint sets whose sums are both (sum of primes dividing n (with repetition))/2.

Original entry on oeis.org

4, 9, 16, 25, 30, 36, 49, 64, 70, 72, 81, 84, 100, 120, 121, 144, 169, 196, 225, 240, 256, 270, 280, 286, 288, 289, 308, 324, 336, 361, 378, 400, 440, 441, 480, 484, 495, 525, 528, 529, 540, 576, 594, 625, 630, 646, 648, 672, 676, 728, 729, 750, 756, 784, 800
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jul 20 2010

Keywords

Comments

From Christian N. K. Anderson, Apr 16 2013: (Start)
Alternatively, the two sets of prime factors have an equal sum.
Superset of even powers, p^(2*i) where p is a prime number (A056798), and composites thereof. (End)

Examples

			a(1)=4 because 4=2*2 and 2=2, a(2)=9 because 9=3*3 and 3=3, a(3)=16 because 16=2*2*2*2 and 2+2=2+2, a(4)=25 because 25=5*5 and 5=5, a(5)=30 because 30=2*3*5 and 2+3=5.

Links

Crossrefs

Prime divisor variant of A083207.
Cf. A001414, A056798, A027746, A221054, A001414.

Programs

  • Haskell
    a175592 n = a175592_list !! (n-1)
a175592_list = filter (z 0 0 . a027746_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
  • Maple
    Primefactors := proc(n) local F, f, i; F := [];
for f in ifactors(n)[2] do
    for i from 1 to f[2] do
        F := [op(F), f[1]]
od od; F end:
isPrimeZumkeller := proc(n) option remember; local s, p, i, P;
    s := add(Primefactors(n));  # A001414
    if s::odd or s = 0 then return false fi;
    P := mul(1 + x^i, i in Primefactors(n));
    is(0 < coeff(P, x, s/2)) end:
select(isPrimeZumkeller, [seq(1..800)]);   # Peter Luschny, Oct 21 2024

Extensions

Corrected by Christian N. K. Anderson, Apr 16 2013

A384498 Squarefree numbers whose distinct prime factors can be partitioned into two sets with equal sums.

Original entry on oeis.org

1, 30, 70, 286, 646, 1798, 2145, 2310, 2730, 3135, 3526, 3570, 4641, 4845, 5005, 5610, 6006, 6279, 6630, 7198, 7410, 7854, 8778, 8855, 8970, 9177, 10366, 10374, 10626, 10695, 11305, 11571, 11730, 13110, 13485, 13566, 13585, 15470, 16095, 16302, 16422, 16530
Offset: 1

Views

Author

Alois P. Heinz, May 31 2025

Keywords

Examples

			2145 = 3*5*11*13 is a term because it is squarefree and 3+13 = 5+11.
16422 = 2*3*7*17*23 is squarefree and 2+7+17 = 3+23.

Links

Crossrefs

Intersection of A005117 and A221054.
Cf. A071141, A071142, A071312.

Programs

  • Maple
    q:= n-> (l-> {l[.., 2][]} minus {1}={} and (s->
        (m-> m::even and coeff(mul(1+x^j, j=s), x, m/2)>0)
        (add(i, i=s)))({l[.., 1][]}))(ifactors(n)[2]):
select(q, [$1..20000])[];
  • Mathematica
    Join[{1},Select[Range[16600],SquareFreeQ[#]&&MemberQ[Total/@Subsets[First/@FactorInteger[#]],Total[First/@FactorInteger[#]]/2]&]] (* James C. McMahon, Jun 02 2025 *)

A384502 Maximum number of distinct prime factors in an n-digit number, n > 3, where its set of distinct prime factors can be partitioned into two equal-sum subsets, each containing at least two elements.

Original entry on oeis.org

5, 5, 7, 7, 7, 9, 9, 9, 11, 11, 11, 13, 13, 13, 15, 15, 15, 16, 17, 17, 17, 19, 19, 19, 19, 21, 21, 21, 21, 23, 23, 23, 23, 25, 25, 25, 25, 27, 27, 27, 27, 29, 29, 29, 29, 31, 31, 31, 31, 33, 33, 33, 33, 34, 35, 35, 35, 35, 37, 37, 37, 37, 39, 39, 39, 39, 39, 41
Offset: 4

Views

Author

Jean-Marc Rebert, May 31 2025

Keywords

Examples

			a(4) = 5, since 2310 = 2 * 3 * 5 * 7 * 11 is a 4-digit number with omega(2310) = 5, and its prime factors can be split into two equal-sum parts: 2 + 5 + 7 = 3 + 11. No 4-digit number that meets this partitioning criterion has an omega value exceeding 5.

Links

Crossrefs

Cf. A067175, A001221, A221054, A383858.

Formula

a(n) <= (largest m such that A067175(m) <= n).

Extensions

a(11)-a(59) from Sean A. Irvine, Jun 23 2025
More terms from David A. Corneth, Aug 15 2025
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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