Sofia January 28 2020 On OEIS A007559 Ivan N. Ianakiev Abstract We prove that, for every integer n > 2, A007559(n) is a Zumkeller number (A083207). Lemma 1: If z is a Zumkeller number, then 2z is also a Zumkeller number. Proof: Let D_1 and D_2 be the sets of positive divisors of z and 2z, written in increasing order as follows: 1 = d_1 < d_2 < … < d_m. If d_i+1 <= 2d_i for 1 <= i < m, then according to Proposition 17 [1] z is a Zumkeller number. Let z be a Zumkeller number. Since D_2 has the same elements as D_1 plus doubles of some of the elements of D_1, then: 1) the sum of the elements of D_2 is also even, and 2) no elements greater than the elements of D_1 times two are introduced into D_1, in order to obtain D_2. Therefore, 2z is also a Zumkeller number. Theorem: For every integer n > 2, A007559(n) is a Zumkeller number. Proof: Let p_1 and p_m be the smallest and the greatest prime in the prime factorization of A007559(n). Let their exponents be e_1 and e_m. For every n > 1, p_1 = 2. For n > 2, e_m is always 1, since a(n) = a(n-1) * (3n + 1), where 3n + 1 is either: a) a prime greater than the greatest prime in the factorization of a(n-1), or b) a composite number with greatest prime factor smaller than the greatest prime factor of a(n-1), in which case the value of e_m does not change. For n > 3, p_m is such that floor(log_2(p_m)) < e_1 (since 3n + 1 is much more often an even number, incl. a power of two, than a prime). Therefore, there exists a power of p_1 e, such that e < e_1 and p_1^e * p_m^e_m is, according to the findings of T. D. Noe [2], a Primitive Zumkeller number. Then, according to Lemma 1 (above) p_1^e_1 * p_m^e_m is a Zumkeller number. Therefore, according to Corollary 5 [1], for n > 2, we prove that a(n) = p_1^e_1 * p_m^e_m * s, where s is relatively prime to p_1 * p_m, is a Zumkeller number. 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 [2] Sloane, N. J. A. (ed.). "Sequence A180332". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.