A290599 -id:A290599 - cp's OEIS frontend

A290602 Irregular triangle read by rows. T(n, k) gives the period length of the periodic sequence {A290600(n, k)^i}_{i >= A290601(n, k)} (mod A002808(n)), for n >= 1 and k = 1..A290599(n).

1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 2, 1, 1, 4, 2, 2, 1, 1, 2, 1, 1, 3, 3, 2, 1, 1, 6, 6, 4, 2, 1, 2, 1, 4, 1, 1, 1, 1, 1, 1, 1, 6, 1, 3, 1, 2, 1, 1, 1, 6, 1, 3, 4, 2, 1, 1, 4, 1, 4, 2, 2, 1, 4, 6, 2, 1, 3, 6, 2, 1, 3, 10, 5, 10, 10, 2, 1, 1, 5, 5, 10, 5, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1

Offset: 1

Wolfdieter Lang, Aug 30 2017

The length of row n is A290599(n).

See A290601 for the proof that this sequence is defined, and the definition of the type of periodicity (imin,P) with imin = A290601(n, k) and the period length P = T(n, k).

			The irregular triangle T(n, k) begins (N(n) = A002808(n)):
n   N(n) \ k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
1   4         1
2   6         2  1  1
3   8         1  1  1
4   9         1  1
5   10        4  2  1  1  4
6   12        2  2  1  1  2  1  1
7   14        3  3  2  1  1  6  6
8   15        4  2  1  2  1  4
9   16        1  1  1  1  1  1  1
10  18        6  1  3  1  2  1  1  1  6  1  3
11  20        4  2  1  1  4  1  4  2  2  1  4
12  21        6  2  1  3  6  2  1  3
13  22       10  5 10 10  2  1  1  5  5 10  5
14  24        2  2  1  1  2  1  1  1  2  2  1  1  2  2  1
15  25        1  1  1  1
...
T(5, 1) = 4 because A290600(5, 1) = 2, N(5) = A002808(5) = 10, A290601(5, 1) = 1 and {2^i}_{i>=1} (mod 10) == {repeat(2,4,8,6)} with period length 4. This is of the type (1,4).
T(7, 6) = 6 because A290600(7, 6) = 10, N(7) = A002808(7) = 14, A290601(7, 6) = 1 and {10^i}_{i>=1} (mod 14) == {repeat(10, 2, 6, 4, 12, 8)} with period length 4. Type (1,6).
The sequence {A290600(10, 1)^i}_{i >= A290601(10, 1)} (mod A002808(10)) = {2^i}_{i >= 1} (mod 18) is periodic with period length P = T(10, 1) = 6. Namely, {repeat(2, 4, 8, 16, 14, 10)}, of type (1,6).
The periodicity types (imin,P) = (A290601(n, k), A290602(n, k)) begin:
n   N(n) \ k    1     2      3      4     5     6     7     8     9      10    11
1   4         (2,1)
2   6         (1,2) (1,1)  (1,1)
3   8         (3,1) (2,1)  (3,1)
4   9         (2,1) (2,1)
5   10        (1,4) (1,2)  (1,1)  (1,1) (1,4)
6   12        (2,2) (1,2)  (1,1)  (2,1) (1,2) (1,1) (2,1)
7   14        (1,3) (1,3)  (1,2)  (1,1) (1,1) (1,6) (1,6)
8   15        (1,4) (1,2)  (1,1)  (1,2) (1,1) (1,4)
9   16        (4,1) (2,1)  (4,1)  (2,1) (4,1) (2,1) (4,1)
10  18        (1,6) (2,1)  (1,3)  (2,1) (1,2) (1,1) (1,1) (2,1) (1,6)  (2,1) (1,3)
11  20        (2,4) (1,2)  (1,1)  (2,1) (1,4) (2,1) (1,4) (2,2) (1,2)  (1,1) (2,4)
12  21        (1,6) (1,2)  (1,1)  (1,3) (1,6) (1,2) (1,1) (1,3)
13  22       (1,10) (1,5) (1,10) (1,10) (1,2) (1,1) (1,1) (1,5) (1,5) (1,10) (1,5)
...
----------------------------------------------------------------------------------

Cf. A002808, A290599, A290600, A290601.

A290600 Irregular triangle T(n, k) read by rows: positive numbers non-coprime to A002808(n) and smaller than A002808(n), sorted increasingly.

Original entry on oeis.org

2, 2, 3, 4, 2, 4, 6, 3, 6, 2, 4, 5, 6, 8, 2, 3, 4, 6, 8, 9, 10, 2, 4, 6, 7, 8, 10, 12, 3, 5, 6, 9, 10, 12, 2, 4, 6, 8, 10, 12, 14, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 3, 6, 7, 9, 12, 14, 15, 18

Offset: 1

Wolfdieter Lang, Aug 30 2017

The length of row n is A290599(n).

Row n gives the complement of row A038566(A002808(n), k) with respect to [1, 2, ..., A002808(n) - 1].

			The irregular triangle T(n, k) begins (N(n) = A002808(n)):
  n   N(n) \ k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
  1   4         2
  2   6         2  3  4
  3   8         2  4  6
  4   9         3  6
  5   10        2  4  5  6  8
  6   12        2  3  4  6  8  9 10
  7   14        2  4  6  7  8 10 12
  8   15        3  5  6  9 10 12
  9   16        2  4  6  8 10 12 14
  10  18        2  3  4  6  8  9 10 12 14 15 16
  11  20        2  4  5  6  8 10 12 14 15 16 18
  12  21        3  6  7  9 12 14 15 18
  13  22        2  4  6  8 10 11 12 14 16 18 20
  14  24        2  3  4  6  8  9 10 12 14 15 16 18 20 21 22
  15  25        5 10 15 20
  ...

Cf. A002808, A038566, A290599, A290601, A290602.

Table[With[{c = FixedPoint[n + PrimePi@ # + 1 &, n + PrimePi@ n + 1]}, Select[Range[c - 1], ! CoprimeQ[#, c] &]], {n, 12}] // Flatten (* Michael De Vlieger, Sep 03 2017 *)

T(n, k) = k-th entry in the list of increasingly sorted numbers of the set {m = 1..A002808(n)-1: gcd(n, m) not equal to 1}.

A290601 Irregular triangle read by rows: T(n, k) gives the least positive integer imin such that the sequence {A290600(n, k)^i}_{i >= imin} (mod A002808(n)) is periodic.

Original entry on oeis.org

2, 1, 1, 1, 3, 2, 3, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 2, 4, 2, 4, 2, 4, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 3, 1, 1, 3, 2, 3, 1, 1, 3, 2, 1, 3, 2, 2, 2, 2

Offset: 1

Wolfdieter Lang, Aug 30 2017

The length of row n is A290599(n).
The corresponding period lengths are given in A290602(n, k).
Conjecture:There exists always an imin >= 1 such that the power sequence A290600(n, k)^i (mod A002808(n)) is periodic for i >= imin. Otherwise T(n, k) is undefined, and one could use T(n, k) = -1. See below for a proof.
This entry resulted from finding the correct initial exponent i for the power sequence considered in triangle A057593 for composite n and gcd(n, k) not equal to 1.
It is clear that the sequence {A290600(n, k)^i}_{i >= 1} is never congruent to 1 (mod A002808(n)) for k = 1..A290599(n). Proof by contradiction. Therefore one can replace 'least positive integer' with 'least nonnegative integer' in the name of this sequence.
The values of these powers of A290600(n, k) modulo A290599(n) are 0 or any of the row entries of A290600(n, k).
To prove periodicity of {A290600(n, k)^i (mod A290599(n))} of the type (imin,P) (starting at i = imin with period length P) one has to solve, with given K = A290600(n, k) and N = A002808(n), the congruence  K^imin * (K^P - 1) == 0 (mod N).
The above conjecture is trivially true because {K^i}{i >= 0} (mod N) becomes (or is) always periodic (K and N positive integer). This follows from the fact that at most N values are possible, namely {0, 1, ..., N-1} and for i >= N one of these values has to appear for the second time, leading to periodicity. Thus the above sequence where gcd(N, K) is not 1 is also defined. - _Wolfdieter Lang, Sep 05 2017

			The irregular triangle T(n, k) begins (N(n) = A002808(n)):
n   N(n) \ k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
1   4         2
2   6         1  1  1
3   8         3  2  3
4   9         2  2
5   10        1  1  1  1  1
6   12        2  1  1  2  1  1  2
7   14        1  1  1  1  1  1  1
8   15        1  1  1  1  1  1
9   16        4  2  4  2  4  2  4
10  18        1  2  1  2  1  1  1  2  1  2  1
11  20        2  1  1  2  1  2  1  2  1  1  2
12  21        1  1  1  1  1  1  1  1
13  22        1  1  1  1  1  1  1  1  1  1  1
14  24        3  1  2  3  1  1  3  2  3  1  1  3  2  1  3
15  25        2  2  2  2
...
T(4, 1) = 2 because A290600(4, 1) = 3, A002808(4) = 9 and {3^i}_{i>=2} (mod 9) = {repeat(0)}, but 3^0 == 1 (mod 9) and  3^1 == 3 (mod 9).
T(4, 2) = 2 because A290600(4, 2) = 6 and  {6^i}_{i>=2} (mod 9) == {repeat(0)}, and 6^0 (mod 9) = 1, 6^1 (mod 9) = 6.
Example for a proof of periodicity of type (1,6) for K = A290600(10, 1) = 2, N = A002808(10) = 18: 2^imin*(2^P - 1) == 0 (mod 18). gcd(2^imin, 18) = 2, and with imin = 1 one has  2^P == 1 (mod 9). Since gcd(2, 9) = 1, a solution exists (Euler): P = phi(9) = 6.

Cf. A002808, A057593, A290599, A290600, A290602.

cp's OEIS Frontend

A290602 Irregular triangle read by rows. T(n, k) gives the period length of the periodic sequence {A290600(n, k)^i}_{i >= A290601(n, k)} (mod A002808(n)), for n >= 1 and k = 1..A290599(n).

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A290600 Irregular triangle T(n, k) read by rows: positive numbers non-coprime to A002808(n) and smaller than A002808(n), sorted increasingly.

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A290601 Irregular triangle read by rows: T(n, k) gives the least positive integer imin such that the sequence {A290600(n, k)^i}_{i >= imin} (mod A002808(n)) is periodic.

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