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 | References | Similar Articles | Additional Information
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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- F. T. Howard, A property of the Rayleigh function, Proc. Amer. Math. Soc. 24 (1970), 719–723. MR 254286, DOI https://doi.org/10.1090/S0002-9939-1970-0254286-8
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- Nand Kishore, The Rayleigh polynomial, Proc. Amer. Math. Soc. 15 (1964), 911–917. MR 168823, DOI https://doi.org/10.1090/S0002-9939-1964-0168823-2
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Additional Information
Keywords:
Rayleigh function,
Bernoulli numbers,
compositions of integers,
enumerating generating function
Article copyright:
© Copyright 1970
American Mathematical Society