[math-fun] Beastly number problem
A beastly number (A051003) is a number with substring 666. It is easy enough to prove that for each positive integer n, there is an integer k such that kn is beastly. Let f(n) be the smallest such k. Show that f(n) is bounded over all integers n. What is the largest possible value of f(n)?
I'll start the bidding with f(16) = 4163. --Rich ---------------- Quoting David Wilson <davidwwilson@comcast.net>:
A beastly number (A051003) is a number with substring 666.
It is easy enough to prove that for each positive integer n, there is an integer k such that kn is beastly. Let f(n) be the smallest such k.
Show that f(n) is bounded over all integers n. What is the largest possible value of f(n)?
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Fairly clearly, this Smallest Beastly Multiple is a special case of the smallest multiple containing a given substring. Each substring produces a similar problem, and intuitively the maximum possible value is in the neighborhood of 10^(1 + log(L)), where L is the length of the substring. If substrings weren't allowed to begin with 0, we could be more parsimonious with this description. Instead of strings we could just use integers, and the bound would be in the neighborhood of 10N. This of course suggests a meta-problem. For a given "beastly subnumber" N, let B(N) be the maximum value of f[N](n) over all n. We expect B(N) to be around 10N, but define b(N) = B(N)/N. What are the extremal values of b(N)?
participants (3)
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Allan Wechsler -
David Wilson -
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