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

Author

Reinhard Zumkeller, Jul 19 2010

Keywords

• nonn

Links

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

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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).