Sofia

January 31 2020



On the question "Which Highly composite numbers (A002182) are Zumkeller numbers (A083207)?"

Ivan N. Ianakiev



Abstract

We prove that, for every integer n > 7, the highly composite number A002182(n) is a Zumkeller number.


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.

Fact 1:

A002182(8) = 48 and A002182(9) = 60 are Zumkeller numbers, which can be easily checked.

Fact 2

Let p_1^e_1 * p_2^e_2 * … * p_m^e_m be the prime factorization of A002182(n). Then, according to the
finding of M. F. Hasler [3], for n > 8, A002182(n) is divisible by 60. Therefore, in its prime
factorization p_1 = 2, which also applies to A002182(8) (see Fact 1).

Fact 3

For n > 7, in the prime factorization of A002182(n) e_m = 1. If it were not so, we could build a smaller
highly composite number with the same number of divisors, in whose primary factorization e_m = 1.

Theorem:

For every integer n > 7, A002182(n) is a Zumkeller number.

Proof:

For n > 7, p_m is such that floor(log_2(p_m)) <= e_1 (see Fact 2 above). If it were not so, then we could
build a smaller highly composite number with the same number of divisors whose prime factorization would
not include p_m, but would include a smaller prime/s to a greater power/s (see Fact 3 above).

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
finding 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 > 7, 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.
https://oeis.org/A180332

[3] Sloane, N. J. A. (ed.). "Sequence A002182". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
https://oeis.org/A002182