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.