login
Primitive Zumkeller numbers.
10

%I #21 Jun 27 2021 03:39:48

%S 6,20,28,70,88,104,272,304,368,464,496,550,572,650,836,945,1184,1312,

%T 1376,1430,1504,1575,1696,1870,1888,1952,2002,2090,2205,2210,2470,

%U 2530,2584,2990,3128,3190,3230,3410,3465,3496,3770,3944,4030,4070,4095,4216

%N Primitive Zumkeller numbers.

%C 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.

%C 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.

%C 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)).

%H T. D. Noe, <a href="/A180332/b180332.txt">Table of n, a(n) for n = 1..9179</a>

%t 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]];

%t 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 *)

%o (Python)

%o from sympy import divisors

%o from sympy.utilities.iterables import subsets

%o def isz(n): # after Peter Luschny in A083207

%o divs = divisors(n)

%o s = sum(divs)

%o if not (s%2 == 0 and 2*n <= s): return False

%o S = s//2 - n

%o R = [m for m in divs if m <= S]

%o return any(sum(c) == S for c in subsets(R))

%o def ok(n): return isz(n) and not any(isz(d) for d in divisors(n)[:-1])

%o print(list(filter(ok, range(1, 5000)))) # _Michael S. Branicky_, Jun 20 2021

%o (SageMath) # uses[is_Zumkeller from A083207]

%o def is_primitiveZumkeller(n):

%o return (is_Zumkeller(n) and

%o not any(is_Zumkeller(d) for d in divisors(n)[:-1]))

%o print([n for n in (1..4216) if is_primitiveZumkeller(n)]) # _Peter Luschny_, Jun 21 2021

%Y Cf. A083207, A006039.

%K nonn

%O 1,1

%A _T. D. Noe_, Sep 07 2010