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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On certain weighted partitions and finite semisimple rings
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by L. B. Richmond and M. V. Subbarao PDF
Proc. Amer. Math. Soc. 64 (1977), 13-19 Request permission

Abstract:

Let k be a fixed integer $\geqslant 1$ and define ${\tau _k}(n) = {\Sigma _{{d^k}/n}}1$. Thus ${\tau _{1}}(n)$ is the ordinary divisor function and ${\tau _k}(n)$ is the number of kth powers dividing n. We derive the asymptotic behaviour as $n \to \infty$ of ${P_k}(n)$ defined by \[ \sum \limits _{n = 0}^\infty {{P_k}(n){x^n} = \prod \limits _{n = 1}^\infty {{{(1 - {x^n})}^{ - {\tau _k}(n)}}} .} \] Thus ${P_k}(n)$ is the number of partitions of n where we recognize ${\tau _k}(m)$ different colours of the integer m when it occurs as a summand in a partition. The case $k = 2$ is of special interest since the number $f(n)$ of semisimple rings with n elements when $n = q_1^{{l_1}}q_2^{{l_2}} \ldots$ is given by $f(n) = {P_2}({l_1}){P_2}({l_2}) \ldots$.
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 64 (1977), 13-19
  • MSC: Primary 10J20
  • DOI: https://doi.org/10.1090/S0002-9939-1977-0439789-1
  • MathSciNet review: 0439789