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

1,32

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.

Links

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

Formula

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

Example

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,...}.

Prog

(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))

Crossrefs

Moebius transform of A354057 applied to each row.

Cf. A327924.

Sequence in context: A338848 A364045 A166301 * A187081 A369308 A212434

Adjacent sequences: A354056 A354057 A354058 * A354060 A354061 A354062

Keyword

nonn,tabl,changed

Author

Jianing Song, May 16 2022

Status

approved