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

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

0,3

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.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..6299 (rows n = 0..6, flattened)

Eric Weisstein's World of Mathematics, Totative

Example

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;

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 *)

Crossrefs

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

Sequence in context: A320390 A292355 A262181 * A281350 A252158 A285996

Adjacent sequences: A309494 A309495 A309496 * A309498 A309499 A309500

Keyword

sign,look,tabf

Author

Jamie Morken, Aug 05 2019

Status

approved