Min_25 / Du Jiao Ladder

Who This Is For

Use this ladder when direct divisor-side prefix sums feel comfortable, but sum phi or sum mu over one huge n still feels like "the prefix sum itself is hidden behind another formula."

Warm-Up

• quotient set Q_n
• grouped floor-division blocks
• why:
• phi * 1 = id
• mu * 1 = e turn into prefix-sum recurrences

Core

• recover one implicit prefix sum on Q_n
• group equal quotient blocks
• solve sum phi first, keep sum mu as the immediate sibling
• flagship rep: Sum of Totient Function

Stretch

• explain why the first route is Du Jiao-first, not full Min_25
• compare direct sum sigma prefix sums with implicit sum phi / sum mu recurrences
• say when prime-counting or other prime-side tasks should push you into the full Min_25 follow-up

Retrieval Layer

• exact starter -> du-jiao-prefix-phi-mu.cpp
• quick reminder sheet -> Min_25 / Du Jiao hot sheet
• compare point -> Dirichlet Convolution / Prefix Sums Of Number-Theoretic Functions

Repo Anchors

• Sum of Totient Function

Exit Criteria

You are ready to move on when you can:

• derive the recurrence for sum phi
• derive the sibling recurrence for sum mu
• explain why Q_x subseteq Q_n matters
• distinguish block lengths from quotient values in the grouped recurrence
• say when this route is still enough and when the full Min_25 follow-up becomes the right next step

External Practice

• Library Checker - Sum of Totient Function
• CSES - Totient sums
• USACO Guide - Prefix Sums of Number-Theoretic Functions (Part 2)

Tutorial Link

• Min_25 / Du Jiao