A380500 Table T(n,k) = phi(phi(prime(n)^k)), n >= 1, k >= 0, read by upwards antidiagonals, where phi = A000010.

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 8, 6, 4, 1, 4, 12, 40, 18, 8, 1, 4, 40, 84, 200, 54, 16, 1, 8, 48, 440, 588, 1000, 162, 32, 1, 6, 128, 624, 4840, 4116, 5000, 486, 64, 1, 10, 108, 2176, 8112, 53240, 28812, 25000, 1458, 128, 1, 12, 220, 2052, 36992, 105456, 585640, 201684, 125000, 4374, 256

Offset: 1

Author

Michael De Vlieger, Feb 04 2025

Keywords

• nonn
• easy
• tabl

Comments

For n >= 2, k >= 1, T(n,k) is the number of primitive roots of prime(n)^k.

Examples

			Table begins as follows:
n\k  0   1     2      3       4        5          6           7
---------------------------------------------------------------
1:   1   1     1      2       4        8         16          32
2:   1   1     2      6      18       54        162         486
3:   1   2     8     40     200     1000       5000       25000
4:   1   2    12     84     588     4116      28812      201684
5:   1   4    40    440    4840    53240     585640     6442040
6:   1   4    48    624    8112   105456    1370928    17822064
7:   1   8   128   2176   36992   628864   10690688   181741696
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11476 (rows n = 1..150, flattened).

Crossrefs

Cf. A000010, A000040, A001248, A008330, A010554, A046144, A104039, A246655.

Programs

Mathematica

Table[EulerPhi[EulerPhi[Prime[#]^k]] &[n - k + 1], {n, 0, 10}, {k, 0, n}] // Flatten

Formula

T(n,k) = A010554(prime(n)^k) = A046144(prime(n)^k).

T(n,0) = 1.

T(n,1) = phi(prime(n)-1) = A008330(n).

T(n,2) = (prime(n)-1) * phi(prime(n)-1)

= (prime(n)-1)^2 * Product_{q|(prime(n)-1)} 1-1/q, prime q.

= A104039(n).

For k > 1, T(n,k) = prime(n)^(k-2) * A104039(n).

T(n,2) > prime(n) for n > 2.

T(n,k) < prime(n)^k for all n and for k > 0.