OFFSET
1,8
COMMENTS
Row n and Row n' are the same if and only if (Z/nZ)* = (Z/n'Z)*, where (Z/nZ)* is the multiplicative group of integers modulo n.
Given n, T(n,k) only depends on gcd(k,psi(n)). For the truncated version see A354060.
Each column is multiplicative.
LINKS
Jianing Song, Table of n, a(n) for n = 1..5050 (the first 100 ascending diagonals)
FORMULA
If (Z/nZ)* = C_{k_1} X C_{k_2} X ... X C_{k_r}, then T(n,k) = Product_{i=1..r} gcd(k,k_r).
T(p^e,k) = gcd((p-1)*p^(e-1),k) for odd primes p. T(2,k) = 1, T(2^e,k) = 2*gcd(2^(e-2),k) if k is even and 1 if k is odd.
A327924(n,k) = Sum_{q|n} T(n,k) * (Sum_{s|n/q} mu(s)/phi(s*q)).
EXAMPLE
n/k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
4 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
5 1 2 1 4 1 2 1 4 1 2 1 4 1 2 1 4 1 2 1 4
6 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
7 1 2 3 2 1 6 1 2 3 2 1 6 1 2 3 2 1 6 1 2
8 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4
9 1 2 3 2 1 6 1 2 3 2 1 6 1 2 3 2 1 6 1 2
10 1 2 1 4 1 2 1 4 1 2 1 4 1 2 1 4 1 2 1 4
11 1 2 1 2 5 2 1 2 1 10 1 2 1 2 5 2 1 2 1 10
12 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4
13 1 2 3 4 1 6 1 4 3 2 1 12 1 2 3 4 1 6 1 4
14 1 2 3 2 1 6 1 2 3 2 1 6 1 2 3 2 1 6 1 2
15 1 4 1 8 1 4 1 8 1 4 1 8 1 4 1 8 1 4 1 8
16 1 4 1 8 1 4 1 8 1 4 1 8 1 4 1 8 1 4 1 8
17 1 2 1 4 1 2 1 8 1 2 1 4 1 2 1 16 1 2 1 4
18 1 2 3 2 1 6 1 2 3 2 1 6 1 2 3 2 1 6 1 2
19 1 2 3 2 1 6 1 2 9 2 1 6 1 2 3 2 1 18 1 2
20 1 4 1 8 1 4 1 8 1 4 1 8 1 4 1 8 1 4 1 8
PROG
(PARI) T(n, k)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(k, Z[i]))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Jianing Song, May 16 2022
STATUS
approved