Learning pathOriginal reference
Lesson 7 of 8

Choose tables from the workload

Combine prime prefixes, exponent summaries, and divisor tables.

Understand the idea

For prime-count intervals, build flags and a prefix count. For divisor counts of every x ≤ N, let each d contribute to d, 2d, 3d, ... . For the factorization of a huge product, add prime exponents from its inputs instead of constructing the product.

Trace a small example

Eight primes lie through 20 and four through 10, so [11,20] contains four. Combining 12 = 2² × 3 and 18 = 2 × 3² gives 2³ × 3³, whose divisor count is 16.

Conditions and cost

Prime prefixes give O(1) queries after preparation. Visiting all multiples costs O(N log N) by the harmonic sum N/1 + ... + N/N. Build reusable tables once and reset query-specific state.

Prefix sum explorer

Query [l, r] is zero-based and inclusive. P[k] stores the total of the first k values; P[0] is zero.

Input array a

30
11
42
13
54
95

Prefix array P

00
31
42
83
94
145
236
P[5] − P[1] = 14 − (3) = 11

Preprocess once in O(n). Each range sum then takes O(1).

Guided simulation · Change the inputs to test your understanding.

Pause & predict

How do prime exponents combine for a product?

Choose an answer
Try it yourself

Turn the idea into code

With P[x] counting primes ≤ x, write the count in [L,R] for L ≥ 1.

Give me a hint

Remove the prefix ending before L.

Show the explained solution

P[R] − P[L − 1], provided both indices lie within the built table.

Your notes

Ready for the next idea?

Mark complete when you can explain it yourself.

From a problem to a programIdentify the input, output, and constraints, then translate a small plan into C++.Variables, types, and remembered valuesChoose a representation, initialize it, and understand what assignment really changes.Arithmetic without surprisesPredict division and remainder, widen before multiplication, and format decimal answers.Conditions and logical decisionsBuild exclusive categories, combine comparisons, and protect operations with short-circuit logic.Optional: other ways to chooseRecognize switch for exact choices and the conditional operator for choosing a value.For loops: make repetition preciseTrace initialization, condition, body, and update—and avoid an extra iteration.While, do-while, and stopping rulesChoose a loop from its stopping condition and read the exact input format.Break, continue, and early answersDistinguish leaving a loop from skipping one iteration, without losing required input.Five patterns worth understandingUse sums, counts, best-so-far candidates, products, and flags to process a stream.Nested loops: rows and columnsTranslate a shape into two loop roles and count the total work.Take a number apart, one digit at a timeUse remainder and division to process decimal digits, including zero.Multiple cases and a debugging mindsetCombine the patterns, isolate each case, and test the boundaries that can break your idea.Arrays: give every value a placeUnderstand indices, initialization, capacity, active length, and the first safe array loop.Scan, count, search, and compareDistinguish values from positions and preserve the exact tie rule a problem asks for.Change a sequence without losing itUpdate, copy, compare, reverse, and inspect neighbours using the correct bounds.Matrices: think in two coordinatesTraverse tables, total rows and columns, inspect diagonals, and protect grid neighbours.Characters become stringsSeparate characters from numbers, then safely inspect and edit managed text.Read the text the problem actually givesChoose words or full lines, preserve empty cases, and reset only the state that needs it.Scan text: count, filter, transformBuild a result from matching characters without losing leading zeros or skipping shifted text.Join, compare, and find textLearn content comparison, lexicographical order, substring counts, and the not-found sentinel.