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A354057
Square array read by ascending antidiagonals: T(n,k) is the number of solutions to x^k == 1 (mod n).
5
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 4, 3, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 3, 4, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 1, 2, 1, 1, 1, 4, 1, 4, 1, 4, 1, 2, 1, 2, 1, 1, 1
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
k-th column: A060594 (k=2), A060839 (k=3), A073103 (k=4), A319099 (k=5), A319100 (k=6), A319101 (k=7), A247257 (k=8).
Applying Moebius transform to the rows gives A354059.
Applying Moebius transform to the columns gives A354058.
Cf. A327924.
Sequence in context: A059233 A357327 A327924 * A143898 A332636 A353742
KEYWORD
nonn,tabl
AUTHOR
Jianing Song, May 16 2022
STATUS
approved