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.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

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

Views

Author

Jamie Morken, Jul 29 2019

Keywords

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

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 *)

A309497 Irregular triangle read by rows: T(n,k) = A060753(n)*k-A038110(n)*A286941(n,k).

Original entry on oeis.org

0, 1, 2, 1, 11, 2, 1, 8, 7, 14, 13, 4, 27, -18, 1, 4, 23, 26, 13, 32, 19, 22, 41, 44, 31, 18, 37, 24, 27, 46, 33, 36, 23, -6, -3, 16, 19, 38, 41, 12, -1, 2, -11, 8, 11, -2, 17, 4, -9, -6, 13, 16, 3, 22, 9, 12, 31, 34, 53, 8
Offset: 0

Views

Author

Jamie Morken, Aug 05 2019

Keywords

Comments

The sequence is Primorial rows of A308121.
Row n has length A005867(n).
Row n > 1 average value = A060753(n)/2.
Row n > 1 has sum = A002110(n-1)*A038110(n)/2.
First value on row(n) = A161527(n-1).
Last value on row(n) = A038110(n) for n > 2.
For n > 1, A060753(n) = Max(row) + Min(row).
For values x and y on row n > 1 at positions a and b on the row:
x + y = A060753(n), where a = A005867(n-1) - (b-1).
For n > 2 the penultimate value on row A002110(n) is given by
A038110(n)*A000040(n)-A060753(n).
Related identity:
A038110(n)/A038111(n)*(Prime(n)^2) - (A038110(n)/A038111(n)*((A038110(n)*Prime(n) - A060753(n))*Prime(n)/A038110(n))) = 1.

Examples

			The triangle starts:
row1: 0;
row2: 1;
row3: 2, 1;
row4: 11, 2, 1, 8, 7, 14, 13, 4;
row5: 27, -18, 1, 4, 23, 26, 13, 32, 19, 22, 41, 44, 31, 18, 37, 24, 27, 46, 33, 36, 23, -6, -3, 16, 19, 38, 41, 12, -1, 2, -11, 8, 11, -2, 17, 4, -9, -6, 13, 16, 3, 22, 9, 12, 31, 34, 53, 8;

Links

Crossrefs

Cf. A058250, A005867, A002110, A038110, A038111, A060753, A286941, A058262, A161527, A083140, A308121.

Programs

  • Mathematica
    row[0] = 0; row[n_] := -(v = Numerator[Product[1 - 1/Prime[i], {i, 1, n}] / Prime[n]] * Select[Range[(p = Product[Prime[i], {i, 1, n}])], CoprimeQ[p, #] &]) + Denominator[Product[((pr = Prime[i]) - 1)/pr, {i, 1, n}]] * Range[Length[v]]; Table[row[n], {n, 0, 4}] // Flatten (* Amiram Eldar, Aug 10 2019 *)
Showing 1-2 of 2 results.

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