A354059 Square array read by ascending antidiagonals: T(n,k) is the number of elements in the multiplicative group of integers modulo n that have order k.

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 3, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Jianing Song, May 16 2022

Keywords

• nonn
• tabl

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.

For the truncated version see A252911.

Examples

			The 7th, 9th, 14th and 18th rows of A354047 are {1,2,3,2,1,6,1,2,3,2,1,6,...}, so applying the Moebius transform gives {1,1,2,0,0,2,0,0,0,0,0,0,...}.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..5050

Crossrefs

Moebius transform of A354057 applied to each row.

Cf. A327924.

Programs

PARI

b(n,k)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(k, Z[i]));
T(n,k) = sumdiv(k, d, moebius(k/d)*b(n,d))

Formula

A327924(n,k) = Sum_{d|k} T(n,k)/phi(d).