OFFSET
1,1
COMMENTS
Observation: The first 19 terms t have the form t = z * p for some Zumkeller number z and some prime p.
Observation holds for first 59 terms. - Michael S. Branicky, Oct 26 2024
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
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..59 (all terms <= 10^7)
EXAMPLE
70688 = 1504 * 47, 1504 is the only Zumkeller divisor of 70688, but 70688 is not Zumkeller.
MAPLE
# The function 'isZumkeller' is defined in A376880.
zdiv := n -> select(isZumkeller, NumberTheory:-Divisors(n)):
select(n -> nops(zdiv(n)) = 1 and op(zdiv(n)) <> n, [seq(1..2000)]);
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Peter Luschny, Oct 20 2024
EXTENSIONS
a(20) and beyond from Michael S. Branicky, Oct 25 2024
STATUS
approved