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Necklace algebra

The algebra $N(R)$ over a ring $R$ with additive group $R^{\mathbf{N}} = \{ a = (a_1,a_2,\ldots) : a_i \in R \}$ and multiplication given by $$ (a * b)_n = \sum_{i,j : \mathrm{lcm}(i,j) = m} \mathrm{hcf}(i,j) a_i b_j \ . $$ The definition of multiplication generalises the Metropolis–Rota multiplication formula for the necklace polynomials.

For a binomial ring $R$, the necklace algebra $N(R)$ is isomorphic to the universal lambda-ring $\Lambda(R)$ via $$ \prod_{n=1}^\infty \left({ 1 - (-t)^n }\right)^{a_n} \longleftrightarrow ( a_n ) \ , $$ which may be regarded as an abstraction of the Artin–Hasse exponential map.

Necklace polynomial

A polynomial of the form $$ M_n(x) = \frac{1}{n} \sum_{d | n} \mu(d) x^{n/d} \ . $$ Here $\mu$ is the Möbius function. When $x$ is a natural number, $M_n(x)$ counts the number of "necklaces": assignments of $n$ colours to $x$ beads under cyclic symmetry which are "primitive", that is, not the repetition of a proper subsequence of colour assignments.

Metropolis and Rota showed that $$ M_n(xy) = \sum_{[i,j]=n} (i,j) M_i(x) M_j(y) $$ Where $[,]$ denotes least common multiple and $(,)$ highest common factor.

Harmonic number

Commonly, a partial sum of the harmonic series $$ H_n = \sum_{k=1}^n \frac{1}{k} \ . $$ A generalised harmonic number is a partial num of the zeta function $$ H_n^{(s)} = \sum_{k=1}^n \frac{1}{k^s} \ . $$

However, Pomerance has defined a harmonic number to be a natural number $n$ for which the harmonic mean of the divisors of $n$ is an integer; equivalently $\sigma(n)$ divides $n.d(n)$ where $\sigma(n)$ is the sum of the divisors of $n$ and $d(n)$ is the number of divisors: these are also called Øre numbers. The first seven such numbers are $$ 1,\ 6,\ 28,\ 140,\ 270,\ 496,\ 672 \ . $$ An even perfect number is a harmonic number.

References

• Guy, Richard K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers". Unsolved Problems in Number Theory (2nd ed.). New York: Springer-Verlag (1994). pp. 16, 45–53
• Milovanović, Gradimir V., Rassias, Michael Th. (edd.) Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava Springer (2014) ISBN 149390258X
• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, edd. Handbook of number theory I. Dordrecht: Springer-Verlag (2006). ISBN 1-4020-4215-9. Zbl 1151.11300
• Sándor, Jozsef; Crstici, Borislav, edd. Handbook of number theory II. Dordrecht: Kluwer Academic (2004). ISBN 1-4020-2546-7. Zbl 1079.11001
• Wagstaff, Samuel S. The Joy of Factoring Student mathematical library 68 American Mathematical Society (2013) ISBN 1470410486