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A376877
Numbers that have exactly one Zumkeller divisor but are not Zumkeller.
3
18, 100, 196, 968, 1352, 4624, 5776, 6050, 8450, 8464, 13456, 15376, 43808, 53792, 59168, 70688, 89888, 111392, 119072, 256036, 287296, 322624, 341056, 399424, 440896, 506944, 602176, 652864, 678976, 732736, 760384, 817216, 1032256, 2196608, 2402432, 2473088, 2841728
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
Subsequence of A376881.
Sequence in context: A087638 A231144 A259231 * A064604 A359435 A301542
KEYWORD
nonn,changed
AUTHOR
Peter Luschny, Oct 20 2024
EXTENSIONS
a(20) and beyond from Michael S. Branicky, Oct 25 2024
STATUS
approved