Jordan's totient J_k(n) = n^k · Π_{p|n}(1 - 1/p^k). For k=1: J_1(n) = φ(n) (Euler). For k=2: J_2(n) counts pairs (a,b) with 1≤a,b≤n and gcd(a,b,n)=1. Multiplicative: J_k(ab) = J_k(a)·J_k(b) when gcd(a,b)=1.
How to Use
Enter n and k:
- J_k(n): Result
- Formula: Prime decomposition
- Compare: Different k values
Special Cases
k=1: φ(n). k=2: J_2(n)=n²·Π(1-1/p²). J_k(p)=p^k - 1 for prime p. J_k(p^a)=p^(ak) - p^(a-1)k). Divisor sum: Σ_{d|n} J_k(d) = n^k.
Applications
- Counting k-dimensional lattice points visible from origin
- Number-theoretic identities
- Generalized Ramanujan sums
- Multiplicative number theory
Step-by-Step Instructions
- 1Enter n.
- 2Choose k.
- 3Compute J_k(n).
- 4See factorization.
- 5Compare k values.