User:Richard Pinch/sandbox-3

Necklace algebra

The algebra over a ring $R$ with additive group $R^{\mathbf{N}} = \{ a = (a_1,a_2,\ldots) : a_i \in R \}$ and multiplication given by $$ (a * b)_n = \sum_{i,j : \mathrm{lcm}(i,j) = m} \mathrm{hcf}(i,j) a_i b_j \ . $$

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