A179528 -id:A179528 - cp's OEIS frontend

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

Reinhard Zumkeller, Jul 19 2010

nonn

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

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

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 *)

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).

A179529 Number of terms of A083207 in 12 consecutive numbers.

Original entry on oeis.org

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

Reinhard Zumkeller, Jul 19 2010

nonn

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).

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

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 *)

A179530 Smallest of 12 consecutive numbers containing exactly n terms of A083207.

Original entry on oeis.org

7, 1, 17, 19, 489, 116859

Offset: 1

Reinhard Zumkeller, Jul 19 2010

A179529(a(n)) = n and A179529(m) <> n for m < a(n);
SUM(A179527(k): a(n) <= k < a(n)+12) = n;
by definition the sequence is finite with at most 12 terms.

			a(3)=17: A179529(17) = #{20,24,28} = #{A083207(3),A083207(4),A083207(5)} = 3;
a(4)=19: A179529(19) = #{20,24,28,30} = #{A083207(3),A083207(4),A083207(5),A083207(6)} = 4;
a(5)=489: A179529(489) = #{490,492,496,498,500} = #{A083207(107),A083207(108),A083207(109),A083207(110),A083207(111)} = 5.
For a(6), the six Zumkeller numbers are 116860, 116862, 116864, 116865, 116868, and 116870. [From _T. D. Noe_, Aug 22 2010]

Peter Luschny, Zumkeller Numbers

Cf. A179528.

a(6) from T. D. Noe, Aug 22 2010

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A179527 Characteristic function of Zumkeller numbers (A083207).

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A179529 Number of terms of A083207 in 12 consecutive numbers.

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A179530 Smallest of 12 consecutive numbers containing exactly n terms of A083207.

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