Journal of Integer Sequences, Vol. 14 (2011), Article 11.2.8

Binomial Coefficient Predictors

Vladimir Shevelev
Department of Mathematics
Ben-Gurion University of the Negev
Beer-Sheva 84105


For a prime $ p$ and nonnegative integers $ n,k,$ consider the set $ A_{n, k}^{(p)}=\{x\in [0,1,...,n]: p^k\vert\vert\binom {n} {x}\}.$ Let the expansion of $ n+1$ in base $ p$ be $ n+1=\alpha_{0}
p^{\nu}+\alpha_{1}p^{\nu-1}+\cdots+\alpha_{\nu},$ where $ 0\leq
\alpha_{i}\leq p-1, i=0, \ldots, \nu.$ Then $ n$ is called a binomial coefficient predictor in base $ p$ ($ p$-BCP), if $ \vert A_{n, k}^{(p)}\vert=\alpha_{k}p^{\nu-k},
k=0,1, \ldots, \nu.$ We give a full description of the $ p$-BCP's in every base $ p.$

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(Concerned with sequence A089633.)

Received March 29 2010; revised version received April 5 2010; September 3 2010; January 19 2011; February 13 2011. Published in Journal of Integer Sequences, March 25 2011.

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