A179529 Number of terms of A083207 in 12 consecutive numbers.

2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 3

Offset: 1

Author

Reinhard Zumkeller, Jul 19 2010

Keywords

• nonn

Comments

a(n) = SUM(A179527(k): n <= k < n+12);

F. Buss and T. D. Noe conjectured a(n) > 0; this is correct (proof by R. Gerbicz);

a(n+1) = A179528(n+12) - A179528(n);

a(A179530(n)) = n and a(m) <> n for m < A179530(n).

Links

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

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_] := Sum[Boole[ZumkellerQ[k]], {k, n, n + 11}];
Array[a, 105] (* Jean-François Alcover, Apr 30 2017, after T. D. Noe *)