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Numbers that have Zumkeller divisors.
5

%I #17 Oct 23 2024 09:26:47

%S 6,12,18,20,24,28,30,36,40,42,48,54,56,60,66,70,72,78,80,84,88,90,96,

%T 100,102,104,108,112,114,120,126,132,138,140,144,150,156,160,162,168,

%U 174,176,180,186,192,196,198,200,204,208,210,216,220,222,224,228,234,240,246,252,258,260,264,270

%N Numbers that have Zumkeller divisors.

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

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

%H Peter Luschny, <a href="/A376880/b376880.txt">Table of n, a(n) for n = 1..10000</a>

%F All terms are even (by transitivity of divisibility).

%e The Zumkeller divisors of 80 are {20, 40, 80}, so 80 is a term.

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

%p with(NumberTheory):

%p isZumkeller := proc(n) option remember; local s, p, i, P;

%p s := SumOfDivisors(n);

%p if s::odd or s < n*2 then false else

%p P := mul(1 + x^i, i in Divisors(n));

%p is(0 < coeff(P, x, s/2)) fi end:

%p select(n -> ormap(isZumkeller, Divisors(n)), [seq(1..270)]);

%t 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 *)

%Y Cf. A083207, A023196, A171641, A376881, A376882.

%K nonn

%O 1,1

%A _Peter Luschny_, Oct 20 2024