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

1,1

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. [See A378538, A378656, A378657].

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)).

The odd terms are not the same as A006038. For example, 342225 occurs there, but not here, while 4448925 occurs here, but is not in A006038. - Antti Karttunen, Dec 05 2024

Links

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

Index entries for sequences where odd perfect numbers must occur, if they exist at all

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 *)

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. A000396 (subsequence), A006038, A083207, A006039, A378537 (characteristic function), A378538, A378656, A378657.

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