A180332 Primitive Zumkeller numbers.
6, 20, 28, 70, 88, 104, 272, 304, 368, 464, 496, 550, 572, 650, 836, 945, 1184, 1312, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002, 2090, 2205, 2210, 2470, 2530, 2584, 2990, 3128, 3190, 3230, 3410, 3465, 3496, 3770, 3944, 4030, 4070, 4095, 4216
Offset: 1
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Mathematica
ZumkellerQ[n_] := ZumkellerQ[n] = Module[{d = Divisors[n], ds, x}, ds = Total[d]; If[OddQ[ds], False, SeriesCoefficient[Product[1 + x^i, {i, d}], {x, 0, ds/2}] > 0]]; Reap[For[n = 1, n <= 5000, n++, If[ZumkellerQ[n] && NoneTrue[Most[Divisors[ n]], ZumkellerQ], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Mar 01 2019 *) -
Python
from sympy import divisors from sympy.utilities.iterables import subsets def isz(n): # after Peter Luschny in A083207 divs = divisors(n) s = sum(divs) if not (s%2 == 0 and 2*n <= s): return False S = s//2 - n R = [m for m in divs if m <= S] return any(sum(c) == S for c in subsets(R)) def ok(n): return isz(n) and not any(isz(d) for d in divisors(n)[:-1]) print(list(filter(ok, range(1, 5000)))) # Michael S. Branicky, Jun 20 2021
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SageMath
# uses[is_Zumkeller from A083207] def is_primitiveZumkeller(n): return (is_Zumkeller(n) and not any(is_Zumkeller(d) for d in divisors(n)[:-1])) print([n for n in (1..4216) if is_primitiveZumkeller(n)]) # Peter Luschny, Jun 21 2021
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