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