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A180332 Primitive Zumkeller numbers.

%I A180332 #28 Dec 05 2024 21:49:33
%S A180332 6,20,28,70,88,104,272,304,368,464,496,550,572,650,836,945,1184,1312,
%T A180332 1376,1430,1504,1575,1696,1870,1888,1952,2002,2090,2205,2210,2470,
%U A180332 2530,2584,2990,3128,3190,3230,3410,3465,3496,3770,3944,4030,4070,4095,4216
%N A180332 Primitive Zumkeller numbers.
%C A180332 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].
%C A180332 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 A180332 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)).
%C A180332 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
%H A180332 T. D. Noe, <a href="/A180332/b180332.txt">Table of n, a(n) for n = 1..9179</a>
%H A180332 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%t A180332 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 A180332 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 A180332 (Python)
%o A180332 from sympy import divisors
%o A180332 from sympy.utilities.iterables import subsets
%o A180332 def isz(n): # after Peter Luschny in A083207
%o A180332     divs = divisors(n)
%o A180332     s = sum(divs)
%o A180332     if not (s%2 == 0 and 2*n <= s): return False
%o A180332     S = s//2 - n
%o A180332     R = [m for m in divs if m <= S]
%o A180332     return any(sum(c) == S for c in subsets(R))
%o A180332 def ok(n): return isz(n) and not any(isz(d) for d in divisors(n)[:-1])
%o A180332 print(list(filter(ok, range(1, 5000)))) # _Michael S. Branicky_, Jun 20 2021
%o A180332 (SageMath) # uses[is_Zumkeller from A083207]
%o A180332 def is_primitiveZumkeller(n):
%o A180332     return (is_Zumkeller(n) and
%o A180332         not any(is_Zumkeller(d) for d in divisors(n)[:-1]))
%o A180332 print([n for n in (1..4216) if is_primitiveZumkeller(n)]) # _Peter Luschny_, Jun 21 2021
%Y A180332 Cf. A000396 (subsequence), A006038, A083207, A006039, A378537 (characteristic function), A378538, A378656, A378657.
%K A180332 nonn
%O A180332 1,1
%A A180332 _T. D. Noe_, Sep 07 2010