A246198 -id:A246198 - cp's OEIS frontend

A331371 Numbers k such that k and k+1 are both half-Zumkeller numbers (A246198).

224, 440, 1224, 2024, 3968, 5624, 11024, 18224, 35720, 38024, 50624, 53360, 65024, 74528, 81224, 140624, 148224, 159200, 164024, 184040, 189224, 194480, 207024, 216224, 233288, 245024, 314720, 354024, 370880, 378224, 416024, 423800, 442224, 455624, 497024, 511224

Offset: 1

Amiram Eldar, May 03 2020

nonn

			224 is a term since both 224 and 225 are half-Zumkeller numbers: the proper divisors of 224 are {1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 112} and 1 + 2 + 4 + 7 + 8 + 14 + 16 + 32 + 56 = 28 + 112, and the proper divisors of 225 are {1, 3, 5, 9, 15, 25, 45, 75} and 1 + 3 + 15 + 25 + 45 = 5 + 9 + 75.

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

Cf. A246198, A246199, A328327.

hzQ[n_] := Module[{d = Most @ Divisors[n], sum, x}, sum = Plus @@ d; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; hzq1 = False; s = {}; Do[hzq2 = hzQ[n]; If[hzq1 && hzq2, AppendTo[s, n - 1]]; hzq1 = hzq2, {n, 2, 6000}]; s

A246199 Odd half-Zumkeller numbers.

Original entry on oeis.org

225, 441, 1225, 2025, 3969, 5625, 11025, 18225, 21609, 27225, 35721, 38025, 50625, 53361, 65025, 74529, 81225, 99225, 119025, 127449, 140625, 148225, 159201, 164025, 184041, 189225, 194481, 207025, 216225, 233289, 245025, 275625, 308025, 314721, 321489

Offset: 1

Chai Wah Wu, Aug 21 2014

nonn

Zumkeller numbers are numbers whose positive divisors can be partitioned into two disjoint sets whose sums are equal (A083207). Half-Zumkeller numbers are numbers whose proper positive divisors can be partitioned into two disjoint sets whose sums are equal (A246198). All numbers in the sequence are not Zumkeller numbers. This is easily seen as the sum of proper divisors is even to be half-Zumkeller, and therefore the sum of the divisors must be odd and thus not Zumkeller.

S. Clark et al., Zumkeller numbers, Mathematical Abundance Conference, April 2008.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..503 from Chai Wah Wu)
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.

Odd numbers in A246198.

Cf. A083207.

from sympy import divisors
import numpy as np
A246199 = []
for n in range(3, 10**5, 2):
    d = divisors(n)
    d.remove(n)
    s, dmax = sum(d), max(d)
    if not s % 2 and 2*dmax <= s:
        d.remove(dmax)
        s2, ld = int(s/2-dmax), 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:
                A246199.append(n)
                break

A290467 Unitary half-Zumkeller numbers: numbers k whose unitary proper divisors can be partitioned into two disjoint sets whose sums are equal.

Original entry on oeis.org

6, 12, 20, 30, 42, 56, 60, 66, 70, 72, 78, 84, 90, 102, 114, 120, 138, 150, 168, 174, 180, 186, 210, 220, 222, 240, 246, 252, 258, 272, 280, 282, 294, 318, 330, 354, 360, 364, 366, 390, 402, 420, 426, 438, 440, 462, 474, 498, 510, 520, 532, 534, 546, 560, 570, 582, 606, 618

Offset: 1

Ivan N. Ianakiev, Aug 03 2017

nonn

Unitary divisors of n are divisors d such that gcd(d,n/d)=1.
Seemingly, a subsequence of A246198 (half-Zumkeller numbers).
The conjecture above is false, since 72, 3600 and 19600 do not belong to A246198. - Ivan N. Ianakiev, Jan 08 2025

			The set of unitary proper divisors of 12 is {1,3,4}. It can be partitioned into two disjoint subsets with equal sums of elements: {1,3} and {4}, therefore 12 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bhabesh Das, On unitary Zumkeller numbers, Notes on Number Theory and Discrete Mathematics, Vol. 30, No. 2 (2024), pp. 436-442.
Eric Weisstein's World of Mathematics, Unitary Divisor Function.
Wikipedia, Unitary divisor.

Cf. A083207, A246198, A290466.

uPropDiv[n_/;n>1]:=Block[{d=Most[Divisors[n]]},Select[d,GCD[#,n/#]==1&]];uhZNQ[n_]:=Module[{d=uPropDiv[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[10^3],uhZNQ] (* combined from the code by Robert G. Wilson v at A034448 and T. D. Noe at A083207 *)

A322658 Integers whose set of proper divisors, excluding 1, can be partitioned into two nonempty subsets having equal sum.

Original entry on oeis.org

36, 72, 105, 144, 195, 200, 255, 288, 315, 324, 345, 385, 392, 400, 450, 495, 525, 576, 585, 648, 675, 735, 784, 800, 805, 825, 855, 882, 900, 945, 975, 1035, 1152, 1155, 1295, 1296, 1305, 1323, 1365, 1395, 1425, 1449, 1463, 1485, 1547, 1568, 1575, 1600, 1645, 1665, 1755, 1764, 1785

Offset: 1

Michel Marcus, Dec 22 2018

nonn

Called half-layered numbers in Behzadipour link.

			36 is a term with {2, 3, 4, 18} and B = {6, 9, 12} having equal sums 27.

Alois P. Heinz, Table of n, a(n) for n = 1..2000
Hussein Behzadipour, Two-layered numbers, arXiv:1812.07233 [math.NT], 2018.

Cf. A083207, A246198, A322657.

a:= proc(n) option remember; local k, l, t, b; b:=
      proc(m, i) option remember; m=0 or i>0 and
        (b(m, i-1) or l[i]<=m and b(m-l[i], i-1)) end;
      for k from 1+`if`(n=1, 1, a(n-1)) do
        if isprime(k) then next fi;
        l:= sort([(numtheory[divisors](k) minus {1, k})[]]);
        t:= add(i, i=l);
        if t::even then forget(b);
          if b(t/2, nops(l)) then return k fi
        fi
      od
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Dec 22 2018

aQ[n_] := CompositeQ[n] && Module[{d = Rest[Most[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[2, 1785], aQ]  (* Amiram Eldar, Dec 22 2018 after T. D. Noe at A083207 *)

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), dd = select(x->((x>1) && (xA083207

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A331371 Numbers k such that k and k+1 are both half-Zumkeller numbers (A246198).

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A246199 Odd half-Zumkeller numbers.

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A290467 Unitary half-Zumkeller numbers: numbers k whose unitary proper divisors can be partitioned into two disjoint sets whose sums are equal.

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A322658 Integers whose set of proper divisors, excluding 1, can be partitioned into two nonempty subsets having equal sum.

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