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A376877 Numbers that have exactly one Zumkeller divisor but are not Zumkeller.

%I A376877 #23 Nov 01 2024 02:53:01
%S A376877 18,100,196,968,1352,4624,5776,6050,8450,8464,13456,15376,43808,53792,
%T A376877 59168,70688,89888,111392,119072,256036,287296,322624,341056,399424,
%U A376877 440896,506944,602176,652864,678976,732736,760384,817216,1032256,2196608,2402432,2473088,2841728
%N A376877 Numbers that have exactly one Zumkeller divisor but are not Zumkeller.
%C A376877 Observation: The first 19 terms t have the form t = z * p for some Zumkeller number z and some prime p.
%C A376877 Observation holds for first 59 terms. - _Michael S. Branicky_, Oct 26 2024
%C A376877 Let us take the respective Zumkeller divisor z of t and find its prime factorization. According to Proposition 2 of Rao/Peng JNT article (see A083207) z has at least one odd prime factor p to an odd power k. By multiplying z by p we make the respective power k in the prime factorization of t even. Therefore, if t is a product of a power of 2 and an even power of an odd prime, then t = z*p. This is the case with 53 of the present 59 terms. - _Ivan N. Ianakiev_, Oct 29 2024
%H A376877 Michael S. Branicky, <a href="/A376877/b376877.txt">Table of n, a(n) for n = 1..59</a> (all terms <= 10^7)
%e A376877 70688 = 1504 * 47, 1504 is the only Zumkeller divisor of 70688, but 70688 is not Zumkeller.
%p A376877 # The function 'isZumkeller' is defined in A376880.
%p A376877 zdiv := n -> select(isZumkeller, NumberTheory:-Divisors(n)):
%p A376877 select(n -> nops(zdiv(n)) = 1 and op(zdiv(n)) <> n, [seq(1..2000)]);
%Y A376877 Subsequence of A376881.
%Y A376877 Cf. A376880, A083207, A376874.
%K A376877 nonn
%O A376877 1,1
%A A376877 _Peter Luschny_, Oct 20 2024
%E A376877 a(20) and beyond from _Michael S. Branicky_, Oct 25 2024