A083207 -id:A083207 - cp's OEIS frontend

A179527 Characteristic function of Zumkeller numbers (A083207).

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0

Offset: 1

Reinhard Zumkeller, Jul 19 2010

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Peter Luschny, Zumkeller Numbers.
Index entries for characteristic functions.

Cf. A057427, A083206, A083207, A179528, A179529.

ZumkellerQ[n_] := Module[{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]];
a[n_] := Boole[ZumkellerQ[n]];
Array[a, 105] (* Jean-François Alcover, Apr 30 2017, after T. D. Noe *)

a(n) = A179528(n+1) - A179528(n).
a(A083207(n)) = 1; a(A083210(n)) = 0.
a(n) = A057427(A083206(n)).
Let n such that a(n)=1 and m coprime to n, then a(m*n)=1, this was proved by R. Gerbicz (lemma for proving A179529(n)>0).

A179529 Number of terms of A083207 in 12 consecutive numbers.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 3

Offset: 1

Reinhard Zumkeller, Jul 19 2010

nonn

a(n) = SUM(A179527(k): n <= k < n+12);
F. Buss and T. D. Noe conjectured a(n) > 0; this is correct (proof by R. Gerbicz);
a(n+1) = A179528(n+12) - A179528(n);
a(A179530(n)) = n and a(m) <> n for m < A179530(n).

R. Zumkeller, Table of n, a(n) for n = 1..10000
Peter Luschny, Zumkeller Numbers

ZumkellerQ[n_] := Module[{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]];
a[n_] := Sum[Boole[ZumkellerQ[k]], {k, n, n + 11}];
Array[a, 105] (* Jean-François Alcover, Apr 30 2017, after T. D. Noe *)

A179528 Number of terms of A083207 that are not greater than n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17

Offset: 1

Reinhard Zumkeller, Jul 19 2010

nonn

Partial sums of A179527: a(n) = SUM(A179527(k): 1<=k<=n);

A179529(n+1) = a(n+12) - a(n).

			a(100)=#{6,12,20,24,28,30,40,42,48,54,56,60,66,70,78,80,84,88,90,96}=20;
a(1000000)=229026, by _T. D. Noe_.

R. Zumkeller, Table of n, a(n) for n = 1..10000
Peter Luschny, Zumkeller Numbers

ZumkellerQ[n_] := Module[{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]];
b[n_] := Boole[ZumkellerQ[n]];
Array[b, 100] // Accumulate (* Jean-François Alcover, Apr 30 2017, after T. D. Noe *)

A328327 Numbers k such that both k and k+1 are Zumkeller numbers (A083207).

Original entry on oeis.org

5984, 7424, 21735, 21944, 26144, 27404, 39375, 43064, 49664, 56924, 58695, 61424, 69615, 70784, 76544, 77175, 79695, 81080, 81675, 82004, 84524, 84644, 89775, 91664, 98175, 103455, 104895, 106784, 109395, 111824, 116655, 116864, 120015, 121904, 122264, 126224

Offset: 1

Amiram Eldar, Oct 12 2019

nonn

Terms k such that both k and k+1 are primitive Zumkeller numbers (A180332) are 82004, 84524, 158235, 516704, 2921535, 5801984, ... (A361934).

There are infinitely many such k as proven by Somu et al. (2023). - Duc Van Khanh Tran, Dec 07 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Giovanni Resta)
Sai Teja Somu, Andrzej Kukla, and Duc Van Khanh Tran, Some results on Zumkeller numbers, arXiv:2310.14149 [math.NT], 2023.

Cf. A083207, A180332, A361934.

Subsequence of A096399.

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]]; zq1 = False; s = {}; Do[zq2 = zumkellerQ[n]; If[zq1 && zq2, AppendTo[s, n - 1]]; zq1 = zq2, {n, 2, 10^5}]; s (* after T. D. Noe at A083207 *)

from itertools import count, islice
from sympy import divisors
def A328327_gen(startvalue=1): # generator of terms >= startvalue
    m = -1
    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:
                    if m == n-1:
                        yield m
                    m = n
                    break
A328327_list = list(islice(A328327_gen(),5)) # Chai Wah Wu, Feb 13 2023

A353061 Zumkeller numbers (A083207) that are not practical numbers (A005153).

Original entry on oeis.org

70, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 350, 354, 366, 372, 402, 426, 438, 444, 474, 490, 492, 498, 516, 534, 550, 564, 572, 582, 606, 618, 636, 642, 650, 654, 678, 708, 732, 762, 770, 786, 804, 822, 834, 836, 852, 876, 894, 906, 910, 940, 942, 945, 948, 978, 996

Offset: 1

Jianing Song, Apr 20 2022

nonn

Different from A007621: A007621 contains no odd numbers, while every odd term in A083207 is here. The numbers 738, 748, 774, 846, ... are in A007621 and are not here.
But the subsequence of even terms (A005843 intersect this sequence) is a subsequence of A007621:
- A007621 = A173490 \ A007620;
- A005843 intersect this sequence = (A005843 intersect A083207) \ A005153;
- A083207 is a subsequence of A023196, and every perfect number is practical;
- So, (A005843 intersect A083207) \ A005153 is a subsequence of A173490, and A005153 is a supersequence of A007620.

			70 is a term since 70 is a Zumkeller number but not a practical number: 1+5+7+10+14+35 = 2+70, so 70 is a Zumkeller number; but 4 cannot be written as a sum of distinct divisors of 70, so 70 is not practical.

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

Cf. A083207, A005153, A007621.

Cf. also A173490, A007620, A023196, A005843.

isA353061(n) = is(n) && !is_A005153(n) \\ See A083207 for is(n) and A005153 for is_A005153(n)

A083208 a(n) = A083206(A083207(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 2, 2, 5, 2, 1, 17, 2, 1, 2, 3, 14, 1, 13, 11, 2, 1, 8, 2, 2, 140, 12, 8, 2, 7, 10, 5, 6, 97, 2, 1, 375, 2, 21, 8, 2, 1, 83, 85, 6, 2, 5, 2, 5, 1090, 2, 273, 2, 4, 58, 73, 1, 2, 59, 2, 4, 209, 1, 4, 4, 53, 2, 13, 56, 781, 4, 4, 2, 3, 3, 2, 11753, 4, 2, 1, 2, 58, 4, 43, 46, 169

Offset: 1

Reinhard Zumkeller, Apr 22 2003

nonn

T. D. Noe, Table of n, a(n) for n=1..10000

A334166 Numbers k having a divisor d, such that d*k is a Zumkeller number (A083207).

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 24, 28, 30, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 92, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 116, 120, 124, 126, 130, 132, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 168, 170, 174, 176, 180, 182, 184, 186, 190

Offset: 1

Ivan N. Ianakiev, Apr 17 2020

nonn

Conjecture: The difference between two consecutive terms is 6 at most.

			2 is a divisor of 10 and 10 is not a Zumkeller number, but 2*10 = 20 is a Zumkeller number, therefore 10 is in the sequence.

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

Supersequence of A083207.

zQ[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]]; fQ[n_]:=AnyTrue[n*Divisors[n],zQ];
Select[Range[200],fQ] (* zQ defined by T. D. Noe at A083207 *)

A360561 a(n) is the least multiple of n that is a Zumkeller number (A083207).

Original entry on oeis.org

6, 6, 6, 12, 20, 6, 28, 24, 54, 20, 66, 12, 78, 28, 30, 48, 102, 54, 114, 20, 42, 66, 138, 24, 150, 78, 54, 28, 174, 30, 186, 96, 66, 102, 70, 108, 222, 114, 78, 40, 246, 42, 258, 88, 90, 138, 282, 48, 294, 150, 102, 104, 318, 54, 220, 56, 114, 174, 354, 60

Offset: 1

Rémy Sigrist, Feb 11 2023

nonn

This sequence is well defined: as stated in Rao and Peng: 6 = 2*3 is a Zumkeller number, so, for any u, v >= 0, 2^(1+2*u) * 3^(1+2*v) is a Zumkeller number, also, if z is a Zumkeller number and m is coprime to z then z*m is also a Zumkeller number; if n = 2^u * 3^v * m with m coprime to 6, let u' be the least odd number >= u and v' be the least odd number >= v, then k = 2^(u'-u) * 3^(v'-v) is an integer (among {1, 2, 3, 6}), k*n is a Zumkeller number and a(n) <= k.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, arXiv:0912.0052 [math.NT], 2009.

Cf. A083207, A360562.

a(n) = { forstep (m=n, oo, n, if (is(m), return (m))) } \\ see A083207 for the function "is"

a(n) = A360562(n) * n.

a(n) = n iff n belongs to A083207.

A179530 Smallest of 12 consecutive numbers containing exactly n terms of A083207.

Original entry on oeis.org

7, 1, 17, 19, 489, 116859

Offset: 1

Reinhard Zumkeller, Jul 19 2010

A179529(a(n)) = n and A179529(m) <> n for m < a(n);
SUM(A179527(k): a(n) <= k < a(n)+12) = n;
by definition the sequence is finite with at most 12 terms.

			a(3)=17: A179529(17) = #{20,24,28} = #{A083207(3),A083207(4),A083207(5)} = 3;
a(4)=19: A179529(19) = #{20,24,28,30} = #{A083207(3),A083207(4),A083207(5),A083207(6)} = 4;
a(5)=489: A179529(489) = #{490,492,496,498,500} = #{A083207(107),A083207(108),A083207(109),A083207(110),A083207(111)} = 5.
For a(6), the six Zumkeller numbers are 116860, 116862, 116864, 116865, 116868, and 116870. [From _T. D. Noe_, Aug 22 2010]

Peter Luschny, Zumkeller Numbers

Cf. A179528.

a(6) from T. D. Noe, Aug 22 2010

A333232 Terms of A051488 that do not belong to A083207.

Original entry on oeis.org

5865, 7395, 10005, 15045, 28815, 37995, 45645, 50235, 99705, 134895, 170085, 275655, 310845, 347565, 391935, 436305, 470235, 486795, 521985, 530265, 590295, 613785, 627555, 635205, 658155, 662745, 707115, 791265, 797385, 830415, 835635, 873885, 887655, 979455, 994755

Offset: 1

Ivan N. Ianakiev, Mar 12 2020

nonn

Giovanni Resta, Table of n, a(n) for n = 1..10000
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.

Cf. A000010, A051488, A083207.

(* First 200000 terms of A051488 *)
a051488=Select[Range[200000],EulerPhi[#]T. D. Noe at A083207 *)
zQ[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]]; t2=Select[t1,!zQ[#]&]

Terms a(12) and beyond from Giovanni Resta, Mar 12 2020

cp's OEIS Frontend

A179527 Characteristic function of Zumkeller numbers (A083207).

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A179529 Number of terms of A083207 in 12 consecutive numbers.

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A179528 Number of terms of A083207 that are not greater than n.

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A328327 Numbers k such that both k and k+1 are Zumkeller numbers (A083207).

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A353061 Zumkeller numbers (A083207) that are not practical numbers (A005153).

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A083208 a(n) = A083206(A083207(n)).

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A334166 Numbers k having a divisor d, such that d*k is a Zumkeller number (A083207).

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A360561 a(n) is the least multiple of n that is a Zumkeller number (A083207).

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A179530 Smallest of 12 consecutive numbers containing exactly n terms of A083207.

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