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id	title	challengeType	forumTopicId	dashedName
5900f4a81000cf542c50ffbb	Problem 316: Numbers in decimal expansions	5	301972	problem-316-numbers-in-decimal-expansions

--description--

Let p = p_1 p_2 p_3 \ldots be an infinite sequence of random digits, selected from {0,1,2,3,4,5,6,7,8,9} with equal probability.

It can be seen that p corresponds to the real number 0.p_1 p_2 p_3 \ldots.

It can also be seen that choosing a random real number from the interval [0,1) is equivalent to choosing an infinite sequence of random digits selected from {0,1,2,3,4,5,6,7,8,9} with equal probability.

For any positive integer n with d decimal digits, let k be the smallest index such that p_k, p_{k + 1}, \ldots p_{k + d - 1} are the decimal digits of n, in the same order.

Also, let g(n) be the expected value of k; it can be proven that g(n) is always finite and, interestingly, always an integer number.

For example, if n = 535, then

for p = 31415926\mathbf{535}897\ldots, we get k = 9

for p = 35528714365004956000049084876408468\mathbf{535}4\ldots, we get k = 36

etc and we find that g(535) = 1008.

Given that \displaystyle\sum_{n = 2}^{999} g\left(\left\lfloor\frac{{10}^6}{n}\right\rfloor\right) = 27280188, find \displaystyle\sum_{n = 2}^{999\\,999} g\left(\left\lfloor\frac{{10}^{16}}{n}\right\rfloor\right).

Note: \lfloor x\rfloor represents the floor function.

--hints--

numbersInDecimalExpansion() should return 542934735751917760.

assert.strictEqual(numbersInDecimalExpansion(), 542934735751917760);

--seed--

--seed-contents--

function numbersInDecimalExpansion() {

  return true;
}

numbersInDecimalExpansion();

--solutions--

// solution required

1.7 KiB Raw Blame History