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Proceedings of the American Mathematical Society

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

 

 

A combinatorial problem and congruences for the Rayleigh function


Author: F. T. Howard
Journal: Proc. Amer. Math. Soc. 26 (1970), 574-578
MSC: Primary 10.07
DOI: https://doi.org/10.1090/S0002-9939-1970-0266853-6
MathSciNet review: 0266853
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Abstract: Let $ z$ be a positive integer and let $ m$ be the number of nonzero terms in the base 2 expansion of $ z$. Define $ f(z,s)$ as the number of positive integers $ r \leqq z/2$ such that the number of nonzero terms in the base 2 expansion of $ r$ plus the number of nonzero terms in the base 2 expansion of $ z - r$ is equal to $ m + s$. We find formulas for $ f(z,s)$ and show how these formulas can be used in proving congruences for the Rayleigh function.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0266853-6
Keywords: Rayleigh function, Bernoulli numbers, compositions of integers, enumerating generating function
Article copyright: © Copyright 1970 American Mathematical Society