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Unitary divisor

A natural number $d$ is a unitary divisor of a number $n$ if $d$ is a divisor of $n$ and $d$ and $n/d$ are coprime, having no common factor other than 1. Equivalently, $d$ is a unitary divisor of $n$ if and only if every prime factor of $d$ appears to the same power in $d$ as in $n$.

The sum of unitary divisors function is denoted by $\sigma^*(n)$. The sum of the $k$-th powers of the unitary divisors is denoted by $\sigma_k^*(n)$. These functions are multiplicative arithmetic functions of $n$ that are not totally multiplicative. The Dirichlet series generating function is

$$ \sum_{n\ge 1}\sigma_k^*(n) n^{-s} = \frac{\zeta(s)\zeta(s-k)}{\zeta(2s-k)} . $$

The number of unitary divisors of $n$ is $\sigma_0(n) = 2^{\omega(n)}$, where $\omega(n)$ is the number of distinct prime factors of $n$.

A unitary or unitarily perfect number is equal to the sum of its aliquot unitary divisors:equivalen tly, it is n such that $\sigma^*(n) = 2n$. A unitary perfect number must be even and it is conjectured that there are only finitely many such. The five known are

$$ 6 = 2\cdot3,\ 60 = 2^2\cdot3\cdot5,\ 90 = 2\cdot3^3\cdot5,\ 87360 = 2^6\cdot3\cdot5\cdot7\cdot13, $$ and $$ 146361946186458562560000 = 2^{18}\cdot3\cdot5^4\cdot7\cdot11\cdot13\cdot19\cdot37\cdot79\cdot109\cdot157\cdot313\ . $$

References

• Guy, Richard K. Unsolved Problems in Number Theory, Problem Books in Mathematics, 3rd ed. (Springer-Verlag, 2004) p.84, section B3. ISBN 0-387-20860-7 Zbl 1058.11001
• Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. (Dordrecht: Kluwer Academic, 2004) pp. 179–327. ISBN 1-4020-2546-7. Zbl 1079.11001
• Wall, Charles R. "The fifth unitary perfect number", Can. Math. Bull. 18 (1975) 115-122. ISSN 0008-4395. Zbl 0312.10004

Descartes number

A Descartes number is a number which is close to being a perfect number. They are named for René Descartes who observed that the number

$$D= 198585576189 = 3^2⋅7^2⋅11^2⋅13^2⋅22021 $$

would be an odd perfect number if only 22021 were a prime number, since the sum-of-divisors function for $D$ satisfies

$$\sigma(D) = (3^2+3+1)\cdot(7^2+7+1)\cdot(11^2+11+1)\cdot(13^3+13+1)\cdot(22021+1) \ . $$

A Descartes number is defined as an odd number $n = m p$ where $m$ and $p$ are coprime and $2n = \sigma(m)\cdot(p+1)$. The example given is the only one currently known.

If $m$ is an odd almost perfect number, that is, $\sigma(m) = 2m-1$, then $m(2m−1)$ is a Descartes number.

References

• Banks, William D.; Güloğlu, Ahmet M.; Nevans, C. Wesley; Saidak, Filip (2008). "Descartes numbers". In De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (edd). Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006. CRM Proceedings and Lecture Notes 46 (Providence, RI: American Mathematical Society) pp. 167–173. ISBN 978-0-8218-4406-9. Zbl 1186.11004.

Multiplicative sequence

Also m-sequence, a sequence of polynomials associated with a formal group structure. They have application in the cobordism ring in algebraic topology.

Definition

Let $K_n$ be polynomials over a ring $A$ in indeterminates $p_1,\ldots$ weighted so that $p_i$ has weight $i$ (with $p_0=1$) and all the terms in $K_n$ have weight $n$ (so that $K_n$ is a polynomial in $p_1,\ldots,p_n$). The sequence $K_n$ is multiplicative if an identity

$$\sum_i p_i z^i = \sum p'_i z^i \cdot \sum_i p''_i z^i $$

implies

$$\sum_i K_i(p_1,\ldots,p_i) z^i = \sum_j K_j(p'_1,\ldots,p'_j) z^j \cdot \sum_k K_k(p''_1,\ldots,p''_k) z^k . $$ The power series

$$\sum K_n(1,0,\ldots,0) z^n $$

is the characteristic power series of the $K_n$. A multiplicative sequence is determined by its characteristic power series $Q(z)$, and every power series with constant term 1 gives rise to a multiplicative sequence.

To recover a multiplicative sequence from a characteristic power series $Q(z)$ we consider the coefficient of z^j in the product

$$ \prod_{i=1}^m Q(\beta_i z) $$

for any $m>j$. This is symmetric in the $\beta_i$ and homogeneous of weight j: so can be expressed as a polynomial $K_j(p_1,\ldots,p_j)$ in the elementary symmetric functions $p$ of the $\beta$. Then $K_j$ defines a multiplicative sequence.

Examples

As an example, the sequence $K_n = p_n$ is multiplicative and has characteristic power series $1+z$.

Consider the power series

$$ Q(z) = \frac{\sqrt z}{\tanh \sqrt z} = 1 - \sum_{k=1}^\infty (-1)^k \frac{2^{2k}}{(2k)!} B_k z^k $$ where $B_k$ is the $k$-th Bernoulli number. The multiplicative sequence with $Q$ as characteristic power series is denoted $L_j(p_1,\ldots,p_j)$.

The multiplicative sequence with characteristic power series

$$ Q(z) = \frac{2\sqrt z}{\sinh 2\sqrt z} $$

is denoted $A_j(p_1,\ldots,p_j)$.

The multiplicative sequence with characteristic power series

$$Q(z) = \frac{z}{1-\exp(-z)} = 1 + \frac{x}{2} - \sum_{k=1}^\infty (-1)^k \frac{B_k}{(2k)!} z^{2k} $$ is denoted $T_j(p_1,\ldots,p_j)$: the Todd polynomials.

Genus

The genus of a multiplicative sequence is a ring homomorphism, from the cobordism ring of smooth oriented compact manifolds to another ring, usually the ring of rational numbers.

For example, the Todd genus is associated to the Todd polynomials with characteristic power series $$\frac{z}{1-\exp(-z)}$$.

References

• Hirzebruch, Friedrich. "Topological methods in algebraic geometry," Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel. Reprint of the 2nd, corr. print. of the 3rd ed. [1978] (Berlin: Springer-Verlag, 1995). ISBN 3-540-58663-6. Zbl 0843.14009.