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

Author

Peter Luschny, Oct 20 2024

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

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

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

Programs

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 *)

Extensions

Incorrect comment removed by Peter Luschny, Dec 02 2024