A279400 -id:A279400 - cp's OEIS frontend

A279399 Irregular triangle read by rows. Row n gives the primes of the smallest positive restricted residue system modulo A033949(n).

3, 5, 7, 5, 7, 11, 2, 7, 11, 13, 3, 5, 7, 11, 13, 3, 7, 11, 13, 17, 19, 2, 5, 11, 13, 17, 19, 5, 7, 11, 13, 17, 19, 23, 3, 5, 11, 13, 17, 19, 23, 7, 11, 13, 17, 19, 23, 29, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 2, 5, 7, 13, 17, 19, 23, 29, 31, 2, 3, 11, 13, 17, 19, 23, 29, 31, 5, 7, 11, 13, 17, 19, 23, 29, 31, 2, 5, 7, 11, 17, 19, 23, 29, 31, 37, 3, 7, 11, 13, 17, 19, 23, 29, 31, 37

Offset: 1

Wolfdieter Lang, Jan 25 2017

The length of row n is given by A279400(n)
For the restricted residue systems modulo n see A038566. For the primes of A038566 (for n >= 3) see A112484.
The primes of the restricted residue system modulo the (composite) positive numbers without a primitive root, given in A033949, are of interest for the determination of the Dirichlet characters modulo the A033949 numbers. For prime numbers (A000040) or for composite positive numbers that have prime primitive roots (A279398) the Dirichlet characters are determined from those of the prime primitive root.

			The triangle T(n, k) begins (here N = A033949(n)):
n,   N \ k 1  2  3  4  5  6  7  8  9 10 ...
1,   8:    3  5  7
2,  12:    5  7 11
3,  15:    2  7 11 13
4,  16:    3  5  7 11 13
5,  20:    3  7 11 13 17 19
6,  21:    2  5 11 13 17 19
7,  24:    5  7 11 13 17 19 23
8,  28:    3  5 11 13 17 19 23
9,  30:    7 11 13 17 19 23 29
10, 32:    3  5  7 11 13 17 19 23 29 31
11, 33:    2  5  7 13 17 19 23 29 31
12, 35:    2  3 11 13 17 19 23 29 31
13, 36:    5  7 11 13 17 19 23 29 31
14, 39:    2  5  7 11 17 19 23 29 31 37
15, 40:    3  7 11 13 17 19 23 29 31 37
...

Cf. A000040, A033949, A038566, A112484, A279398, A279400, A279401.

Row n of T is given by the primes of row A033949(n) of A038566, for n >= 1.

T(n, k) = A112484(A033949(n), k), n >= 1, k = 1..A279400(n).

A279401 Irregular triangle read by rows. Row n gives the orders of the primes of row n of the irregular triangle A279399 modulo A033949(n).

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 4, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 2, 4, 4, 2, 6, 6, 6, 2, 6, 6, 2, 2, 2, 2, 2, 2, 2, 6, 6, 6, 2, 6, 6, 6, 4, 2, 4, 4, 2, 4, 2, 8, 8, 4, 8, 8, 2, 8, 4, 8, 2, 10, 10, 10, 10, 10, 10, 2, 10, 5, 12, 12, 3, 4, 12, 6, 12, 2, 6, 6, 6, 6, 3, 2, 2, 6, 6, 6, 12, 4, 12, 12, 6, 12, 6, 6, 4, 12, 4, 4, 2, 4, 4, 2, 4, 2, 2, 4

Offset: 1

Wolfdieter Lang, Jan 30 2017

The length of row n is given by A279400(n).
See the A279399 comments.
The entries in row n are proper divisors of phi(A033949(n)), where phi(n) = A000010(n).
  This is because no A033949 number has a primitive root.

			The irregular triangle T(n, k) begins (here N = A033949(n)):
n,   N \ k 1  2  3  4  5  6  7  8  9 10 ...
1,   8:    2  2  2
2,  12:    2  2  2
3,  15:    4  4  2  4
4,  16:    4  4  2  4  4
5,  20:    4  4  2  4  4  2
6,  21:    6  6  6  2  6  6
7,  24:    2  2  2  2  2  2  2
8,  28:    6  6  6  2  6  6  6
9,  30:    4  2  4  4  2  4  2
10, 32:    8  8  4  8  8  2  8  4  8  2
11, 33:   10 10 10 10 10 10  2 10  5
12, 35:   12 12  3  4 12  6 12  2  6
13, 36:    6  6  6  3  2  2  6  6  6
14, 39:   12  4 12 12  6 12  6  6  4 12
15, 40:    4  4  2  4  4  2  4  2  2  4
...
The sequence of phi(N) begins: 4, 4, 8, 8, 8, 12, 8, 12, 8, 16, 20, 24, 12, 24, 16, ...
n = 2, N = 12:  5^2 == 7^2 == 11^2 == 1 (mod 12), therefore 2 is the least positive power k for each of the three primes p of row 2 of A279399 which satisfies p^k == 1 (mod A033949(2)).

Cf. A000010, A033949, A279399, A279400.

T(n, k) = order(A279399(n, k)) (mod A033949(n)), n >= 1, k = 1..A279400(n).

cp's OEIS Frontend

A279399 Irregular triangle read by rows. Row n gives the primes of the smallest positive restricted residue system modulo A033949(n).

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