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

1,1

Comments

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.

Links

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

Example

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.

Maple

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)]);

Mathematica

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 *)

Crossrefs

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

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

Sequence in context: A326133 A177052 A023196 * A204829 A005835 A007620

Adjacent sequences: A376877 A376878 A376879 * A376881 A376882 A376883

Keyword

nonn

Author

Peter Luschny, Oct 20 2024

Extensions

Incorrect comment removed by Peter Luschny, Dec 02 2024

Status

approved