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A179527 Characteristic function of Zumkeller numbers (A083207).

%I A179527 #26 Jul 23 2025 14:36:51
%S A179527 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,
%T A179527 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,
%U A179527 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
%N A179527 Characteristic function of Zumkeller numbers (A083207).
%H A179527 Reinhard Zumkeller, <a href="/A179527/b179527.txt">Table of n, a(n) for n = 1..10000</a>
%H A179527 Peter Luschny, <a href="http://www.luschny.de/math/seq/ZumkellerNumbers.html">Zumkeller Numbers</a>.
%H A179527 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.
%F A179527 a(n) = A179528(n+1) - A179528(n).
%F A179527 a(A083207(n)) = 1; a(A083210(n)) = 0.
%F A179527 a(n) = A057427(A083206(n)).
%F A179527 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).
%t A179527 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]];
%t A179527 a[n_] := Boole[ZumkellerQ[n]];
%t A179527 Array[a, 105] (* _Jean-François Alcover_, Apr 30 2017, after _T. D. Noe_ *)
%o A179527 (Poly/ML)
%o A179527 fun A179527(n) =
%o A179527     let fun ch(m, k) =
%o A179527             if k <= m
%o A179527                then ch(m, k+1) orelse (n mod k = 0 andalso ch(m-k, k+1))
%o A179527                else (m = 0)
%o A179527      in if A000203(n) mod 2 = 0 andalso ch(A000203(n) div 2 - n, 1)
%o A179527            then 1
%o A179527            else 0
%o A179527     end;
%Y A179527 Cf. A057427, A083206, A083207, A179528, A179529.
%K A179527 nonn
%O A179527 1,1
%A A179527 _Reinhard Zumkeller_, Jul 19 2010