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A180332 Primitive Zumkeller numbers. 7
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 (list; graph; refs; listen; history; text; internal format)
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 non-deficient 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]] (* Jean-Fran├žois 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

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Last modified July 6 06:57 EDT 2022. Contains 355108 sequences. (Running on oeis4.)