cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A279400 Row lengths of the irregular triangle A279399.

Original entry on oeis.org

3, 3, 4, 5, 6, 6, 7, 7, 7, 10, 9, 9, 9, 10, 10, 10, 12, 12, 13, 13, 13, 14, 14, 14, 14, 16, 17, 16, 15, 17, 17, 16, 18, 19, 19, 19, 18, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 22, 23, 23
Offset: 1

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Author

Wolfdieter Lang, Jan 25 2017

Keywords

Comments

a(n) is the number of primes of the smallest positive restricted residue system modulo A033949(n).

Crossrefs

Cf. A033949, A279399.

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

Views

Author

Wolfdieter Lang, Jan 30 2017

Keywords

Comments

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.

Examples

			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)).

Crossrefs

Cf. A000010, A033949, A279399, A279400.

Formula

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

A282624 Irregular triangle read by rows: row n gives a certain choice of generators of the multiplicative group of integers modulo A033949(n).

Original entry on oeis.org

3, 5, 5, 7, 2, 11, 3, 7, 3, 11, 2, 13, 5, 7, 13, 3, 13, 7, 11, 3, 31, 2, 23, 19, 13, 5, 19, 17, 5, 3, 11, 29, 5, 13, 3, 43, 11, 17, 5, 7, 17, 5, 35, 3, 5, 19, 23, 3, 13, 29, 2, 37, 7, 11, 19, 2, 5, 3, 31, 2, 31, 5, 43, 3, 67, 2, 68, 19, 13, 5, 17, 19, 11, 7
Offset: 1

Views

Author

Wolfdieter Lang, Mar 03 2017

Keywords

Comments

The length of row n is given by A046072(A033949(n)), n >= 1.
The generators are chosen minimally in the sense that the product of their orders (cycle lengths) is phi(N(n)) = A000010(N(n)) with N(n) = A033949(n). In addition, the generators are sorted with nonincreasing orders, and the smallest numbers with these orders are listed.
Note that the first instance where a composite generator is needed is N = 51 = A033949(20) with a generator 35. The next such number is N = 69 = A033949(31) with a generator 68. Such numbers N will be called exceptional.
For a table with n = 1..69, N = 8, 12, ..., 130, see the W. Lang link. Compare this with the Wikipedia table (where some generator errors will be corrected). There non-minimal generators are also used, i.e., the product of the orders of the generators is larger than phi(N). The Wikipedia table often uses composite generators when primes would do the job. E.g., N = 16 with generators 2, 14 instead of 2, 11; or N = 16 with 3, 15 instead of 3, 7, etc.

Examples

			The irregular triangle T(n, k) begins (here N = A033949(n), and the respective primitive cycle lengths and phi(N) are also given)
n,   N \k 1   2   3 ... cycle lengths, phi(N)
1,   8:   3   5           2  2          4
2,  12:   5   7           2  2          4
3,  15:   2  11           4  2          8
4,  16:   3   7           4  2          8
5,  20:   3  11           4  2          8
6,  21:   2  13           6  2         12
7,  24:   5   7  13       2  2  2       8
8,  28:   3  13           6  2         12
9,  30:   7  11           4  2          8
10, 32:   3  31           8  2         16
11, 33:   2  23          10  2         20
12, 35:  19  13           6  4         24
13, 36:   5  19           6  2         12
14, 39:  17   5           6  4         24
15, 40:   3  11  29       4  2  2      16
16: 42:   5  13           6  2         12
17, 44:   3  43          10  2         20
18, 45:  11  17           6  4         24
19, 48:   5   7  17       4  2  2      16
20, 51:   5  35          16  2         32
... See the link for more.

Links

Crossrefs

Cf. A033949, A046072, A279399, A281855, A281856, A282623, A282625.
Showing 1-3 of 3 results.

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