Sofia
January 5 2020


Proof of a Conjecture Related to OEIS A330585

Ivan N. Ianakiev


Abstract
OEIS A330585 is proved to be a subsequence of OEIS A083207 (aka Zumkeller numbers).


It is easy to check the first 79 terms of A330585 with the pari script provided in A083207.
The proof for the last 3 terms is based on the conclusions by Yuejian Peng and Bhaskara Rao [1].

1. a(80)
Take the Zumkeller number 350 (factorization 2 * 5^2 * 7). According to Proposition 6 [1],
2^21 * 5^2 * 7^3 = 17983078400 is also a Zumkeller number. Therefore, according to Corollary 5 [1], 
a(80) = a*b = 17983078400*69799268026917, where a and b are relatively prime, is a Zumkeller number.

2. a(81)
Take the Zumkeller number 6 (factorization 2^1 * 3^1). According to Proposition 6 [1], the number 
4586471424 (factorization 2^41 * 3^13) is also a Zumkeller number. Therefore, according to Corollary 5 [1],
a(81) = a*b = 4586471424*1185064677671875, where a and b are relatively prime, is a Zumkeller number. 

3. a(82)
According to Proposition 9 [1], 1196268651020288 = 2^46 * 17 is Zumkeller number. Therefore, according to
Corollary 5 [1], a(82) = a*b = = 1196268651020288*675448131241557866586467272529408203125, where a and b
are relatively prime, is a Zumkeller number.

Thus, the fact that A330585 is a subsequence of A083207 has been established.


References

[1] Yuejian Peng, K.P.S. Bhaskara Rao, On Zumkeller numbers,
Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.
https://doi.org/10.1016/j.jnt.2012.09.020