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

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.

Cf. A376880, A083207, A376874.

Sequence in context: A087638 A231144 A259231 * A064604 A359435 A301542

Adjacent sequences: A376874 A376875 A376876 * A376878 A376879 A376880

Keyword

nonn

Author

Peter Luschny, Oct 20 2024

Extensions

a(20) and beyond from Michael S. Branicky, Oct 25 2024

Status

approved