A008330 -id:A008330 - cp's OEIS frontend

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

Michael De Vlieger, Feb 04 2025

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

			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

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

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

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

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.

A067733 Numbers k such that phi(prime(k)-1) == 0 (mod k).

Original entry on oeis.org

1, 12, 540, 1024, 1072, 1092, 1320, 1408, 4272, 16200, 29568, 40082, 43740, 56592, 123192, 251736, 265440, 276000, 664320, 725760, 758560, 771264, 2101248, 3195192, 6592704, 7243050, 7464960, 8755680, 10553502, 11715840, 14135296, 19952730, 20058720, 21672288

Offset: 1

Benoit Cloitre, Feb 05 2002

nonn

Cf. A000010, A006093, A008330.

Select[Range[42000], Mod[EulerPhi[Prime[ # ] - 1], # ] == 0 &]

for(k=1,2600000, if(eulerphi(prime(k)-1)%k==0,print1(k,",")))

More terms from Klaus Brockhaus, Feb 13 2002

More terms from Jinyuan Wang, Apr 05 2020

A306371 Number of primitive roots of prime A103521(n).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 20, 24, 32, 36, 40, 48, 64, 72, 80, 96, 120, 144, 160, 176, 200, 216, 240, 288, 320, 336, 384, 432, 448, 480, 576, 720, 768, 880, 960, 1056, 1200, 1280, 1344, 1440, 1664, 1680, 1728, 1920, 2112, 2208, 2304, 2400, 2592, 2784, 2880, 3072, 3456, 3840, 4224, 4320

Offset: 1

Jianing Song, Feb 11 2019

nonn

Numbers k in A008330 such that no numbers <= k occur later than k in A008330.

Different from A036912 since a(19) = 144 and A036912(19) = 128.

Cf. A103521.

Cf. also A103203, A121519.

b(n) = if(n==1, 2, floor(exp(Euler)*n*log(log(n^2))+2.5*n/log(log(n^2))));
f(p) = my(i=0); forprime(q=p+1, b(eulerphi(p-1))+1, i+=(eulerphi(q-1)<=eulerphi(p-1))); i;
forprime(p=2, 2e4, if(f(p)==0, print1(eulerphi(p-1), ", ")))

a(n) = phi(A103521(n)-1).

A317706 Irregular triangle of numbers k < p^2 such that k is a primitive root modulo p but not p^2, p = prime(n).

Original entry on oeis.org

1, 8, 7, 18, 19, 31, 40, 94, 112, 118, 19, 80, 89, 150, 40, 65, 75, 131, 158, 214, 224, 249, 116, 127, 262, 299, 307, 333, 28, 42, 63, 130, 195, 263, 274, 352, 359, 411, 14, 60, 137, 221, 374, 416, 425, 467, 620, 704, 781, 827, 115, 117, 145, 229, 414, 513, 623, 726

Offset: 1

Jianing Song, Aug 05 2018

Also row n lists numbers k < p^2 such that the multiplicative order of k modulo p^2 is p - 1.
Row n has phi(prime(n) - 1) = A008330(n) terms.
Row sum is congruent to mu(prime(n) - 1) = A089451(n) modulo prime(n)^2, where mu is the Moebius function. For n >= 3, the product of n-th row is congruent to 1 modulo prime(n)^2.
Does every integer appear in this sequence? For example, 3 does not appear until the prime 1006003 and 5 does not appear until the prime 40487. Where does 2 first appear?

			(2)   1,
(3)   8,
(5)   7, 18,
(7)   19, 31,
(11)  40, 94, 112, 118,
(13)  19, 80, 89, 150,
(17)  40, 65, 75, 131, 158, 214, 224, 249,
(19)  116, 127, 262, 299, 307, 333,
(23)  28, 42, 63, 130, 195, 263, 274, 352, 359, 411,

Cf. A008330, A055578, A089451, A143548.

Table[Select[Range[p^2 - 1], MultiplicativeOrder[#, p^2] == p - 1 &], {p, Prime@ Range@ 11}] // Flatten (* Michael De Vlieger, Aug 05 2018 *)

forprime(p=2,100,for(i=1,p^2,if(Mod(i,p)!=0,if(znorder(Mod(i,p^2))==p-1,print1(i, ", ")))))

cp's OEIS Frontend

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

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A067733 Numbers k such that phi(prime(k)-1) == 0 (mod k).

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A306371 Number of primitive roots of prime A103521(n).

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Formula

A317706 Irregular triangle of numbers k < p^2 such that k is a primitive root modulo p but not p^2, p = prime(n).

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