A376879 -id:A376879 - cp's OEIS frontend

A376880 Numbers that have Zumkeller divisors.

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264, 270

Offset: 1

Peter Luschny, Oct 20 2024

nonn

d is a Zumkeller divisor of n if and only if d is a divisor of n and is Zumkeller (A083207).

The first difference from A023196 is 748, which is abundant (sigma(748) = 1512 > 2*748) but has no Zumkeller divisors.

			The Zumkeller divisors of 80 are {20, 40, 80}, so 80 is a term.
The divisors of 81 are {1, 3, 9, 27, 81}, none of which is Zumkeller, so 81 is not a term.

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

Cf. A083207, A023196, A171641, A376879, A376881.

Positions of terms > 1 in A376882, terms > 0 in A378446.

with(NumberTheory):
isZumkeller := proc(n) option remember; local s, p, i, P;
    s := SumOfDivisors(n);
    if s::odd or s < n*2 then false else
    P := mul(1 + x^i, i in Divisors(n));
    is(0 < coeff(P, x, s/2)) fi end:
select(n -> ormap(isZumkeller, Divisors(n)), [seq(1..270)]);

znQ[n_]:=Length[Select[{#, Complement[Divisors[n], #]}&/@Most[Rest[ Subsets[ Divisors[ n]]]], Total[#[[1]]]==Total[#[[2]]]&]]>0; zn=Select[Range[300], znQ] (* zn from A083207 *) ;Select[Range[270],IntersectingQ[Divisors[#],zn]&] (* James C. McMahon, Oct 23 2024 *)

Incorrect comment removed by Peter Luschny, Dec 02 2024

A378656 Numbers that are primitive non-deficient, but not primitive Zumkeller.

Original entry on oeis.org

748, 7544, 10184, 56816, 61904, 62416, 66928, 69488, 73616, 102416, 195316, 292604, 297908, 342225, 394144, 517024, 543968, 640096, 682592, 776096, 955424, 1047392, 1088288, 2081824, 2154584, 2227616, 3239744, 3414848, 3593792, 3839296, 3921856, 3963968, 4082368, 4114624, 4130624, 4147648, 4181696, 4280768, 4315072

Offset: 1

Antti Karttunen, Dec 05 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..50

Sequence A006039 without any terms of A180332.
Setwise difference A378538 \ A378657.
Subsequence of A376879.
Cf. A341619, A378537.

is_A378656(n) = (A341619(n) && !A378537(n));

{k such that A341619(k) = 1 and A378537(k) = 0}.

A378519 Numbers which can be written in precisely one way as sum of a subset of their proper divisors but are not Zumkeller numbers, i.e., have no subsets of their divisors such that the complement has the same sum.

Original entry on oeis.org

748, 7544, 10184, 61904, 66928

Offset: 1

Antti Karttunen, Dec 01 2024

Question: Is this a subsequence of A376879?

Intersection of A064771 and A083210.
Positions of 108's in A378604.
Cf. A376879.

cp's OEIS Frontend

A376880 Numbers that have Zumkeller divisors.

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A378656 Numbers that are primitive non-deficient, but not primitive Zumkeller.

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A378519 Numbers which can be written in precisely one way as sum of a subset of their proper divisors but are not Zumkeller numbers, i.e., have no subsets of their divisors such that the complement has the same sum.

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