A307964 Irregular triangle read by rows: T(n,k) = A308121(A024556(n),k).

7, 14, 13, 4, 11, 2, 1, 8, 3, 6, 5, 8, 3, 2, 5, 4, -1, 2, 1, 4, 13, 26, 19, 32, 25, 38, 31, 4, 17, 10, 23, 16, 29, 2, -5, 8, 1, 14, 7, 20, 11, 22, 33, 44, 31, 18, 29, 16, 27, 38, 1, 12, 23, 34, -3, 8, 19, 6, 17, 4, -9, 2, 13, 24, 5, 10, 7, 12, 9, 14, 11, 16, 5

Offset: 1

Author

Jamie Morken, Jul 29 2019

Keywords

• sign
• look
• tabf

Comments

The sequence gives odd squarefree composite rows n in A308121, i.e., rows 15, 21, 33, 35, 39, 51, 55, 57, 65, ... given by A024556(n). These rows are the primitive rows of A308121.

Row n has length A000010(A024556(n)).

For row n:

T(n, 1) = T(n, 2) / 2.

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

T(n, phi(n)/2+1) - T(n, phi(n)/2) = T(n, 1).

From Charlie Neder, Jul 30 2019: (Start)

For row n, T(n, k) + T(n, phi(n)-k) is constant for all k.

For 2 <= k < lpf(A024556(n)), T(n, k) = k*T(n, 1). (End)

Examples

			The sequence as an irregular triangle:
1:  7, 14, 13, 4, 11, 2, 1, 8;
2:  3, 6, 5, 8, 3, 2, 5, 4, -1, 2, 1, 4;
3:  13, 26, 19, 32, 25, 38, 31, 4, 17, 10, 23, 16, 29, 2, -5, 8, 1, 14, 7, 20;
4:  11, 22, 33, 44, 31, 18, 29, 16, 27, 38, 1, 12, 23, 34, -3, 8, 19, 6, 17, 4, -9, 2, 13, 24;
5:  5, 10, 7, 12, 9, 14, 11, 16, 5, 2, 7, 4, 9, 6, 11, 8, -3, 2, -1, 4, 1, 6, 3, 8
6:  19, 38, 25, 44, 31, 50, 37, 56, 43, 62, 49, 4, 23, 10, 29, 16, 35, 22, 41, 28, 47, 2, -11, 8, -5, 14, 1, 20, 7, 26, 13, 32;
7:  3, 6, 9, 12, 7, 10, 13, 16, 3, 6, 9, 4, 7, 10, 13, 8, 3, 6, 1, 4, 7, 10, 5, 8, 3, -2, 1, 4, 7, 2, 5, 8, -5, -2, 1, 4, -1, 2, 5, 8;
8:  7, 14, 9, 16, 11, 18, 13, 20, 15, 22, 17, 24, 7, 2, 9, 4, 11, 6, 13, 8, 15, 10, 17, 12, -5, 2, -3, 4, -1, 6, 1, 8, 3, 10, 5, 12;
9:  17, 34, 51, 68, 37, 54, 71, 88, 57, 74, 43, 12, 29, 46, 63, 32, 49, 66, 83, 4, 21, 38, 7, 24, 41, 58, 27, 44, 61, -18, -1, 16, 33, 2, 19, 36, 53, 22, -9, 8, -23, -6, 11, 28, -3, 14, 31, 48;
  ...
		

Links

• Jamie Morken, Table of n, a(n) for n = 1..9768

Crossrefs

Cf. A024556, A308121, A309497, A000010, A076511, A076512.

Programs

Mathematica

rowsToCheck = 340;
A024556 =
Complement[Select[Range[3, rowsToCheck, 2], SquareFreeQ],
  Prime[Range[
    PrimePi[rowsToCheck]]]]; (* after Harvey P. Dale , Jan 26 2011 *)
A308121 =
Table[With[{a = n/GCD[n, #], b = Numerator[#/n]},
     MapIndexed[a First@#2 - b #1 &,
      Flatten@Position[GCD[Table[Mod[k, n], {k, n - 1}], n],
         1] /. {} -> {1}]] &@EulerPhi@n, {n,
   rowsToCheck}]; (* after Michael De Vlieger, Jun 06 2019 *)
A307964 = {};
For[i = 1, i <= Length[A024556], i++,
AppendTo[A307964, A308121[[A024556[[i]]]]]]
A307964flattened = Flatten[A307964]
(* Jamie Morken, Apr 20 2021 *)