

A180332


Primitive Zumkeller numbers.


8



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
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OFFSET

1,1


COMMENTS

A number is called a primitive Zumkeller number if it is a Zumkeller number (A083207) but none of its proper divisors are Zumkeller numbers. These numbers are very similar to primitive nondeficient numbers (A006039), but neither is a subsequence of the other.
Because every Zumkeller number has a divisor that is a primitive Zumkeller number, every Zumkeller number z can be factored as z = d*r, where d is the smallest divisor of z that is a primitive Zumkeller number.
Every number of the form p*2^k is a primitive Zumkeller number, where p is an odd prime and k = floor(log_2(p)).


LINKS

T. D. Noe, Table of n, a(n) for n = 1..9179


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]] (* JeanFrançois Alcover, Mar 01 2019 *)


PROG

(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
(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


CROSSREFS

Cf. A083207, A006039.
Sequence in context: A119425 A342669 A006039 * A338133 A064771 A006036
Adjacent sequences: A180329 A180330 A180331 * A180333 A180334 A180335


KEYWORD

nonn


AUTHOR

T. D. Noe, Sep 07 2010


STATUS

approved



