A308121 Irregular triangle read by rows: T(n,k) = A109395(n)*k-A076512(n)*A038566(n,k).

0, 1, 1, 2, 1, 1, 1, 2, 3, 4, 2, 1, 1, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 1, 2, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 1, 2, 3, 7, 14, 13, 4, 11, 2, 1, 8

Offset

1,4

Comments

Row n has length A000010(n).

Row n > 1 has sum = n*A076512(n)/2.

First value on row(n) = A076511(n).

Last value on row(n) = A076512(n) for n > 1.

For n > 1, A109395(n) = Max(row) + Min(row).

For values x and y on row n > 1 at positions a and b on the row:

x + y = A109395(n), where a = A000010(n) - (b-1).

For n > 2 the penultimate value on row A002110(n) is given by

A038110(n)*A000040(n)-A060753(n).

From Charlie Neder, Jun 05 2019: (Start)

If p is a prime dividing n, then row p*n consists of p copies of row n.

Conjecture: If n is odd, then row 2n can be obtained from row n by interchanging the first and second halves. (End)

Links

Jamie Morken, Table of n, a(n) for n = 1..13413 (Rows n = 1..210 of triangle, flattened)

Example

The sequence as an irregular triangle:
  n/k 1, 2, 3, 4, ...
   1: 0
   2: 1
   3: 1, 2
   4: 1, 1
   5: 1, 2, 3, 4
   6: 2, 1
   7: 1, 2, 3, 4, 5, 6
   8: 1, 1, 1, 1
   9: 1, 2, 1, 2, 1, 2
  10: 3, 4, 1, 2
  11: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
  12: 2, 1, 2, 1
  13: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
  14: 4, 5, 6, 1, 2, 3
  15: 7, 14, 13, 4, 11, 2, 1, 8
  ...
  Row sums: 0, 1, 3, 2, 10, 3, 21, 4, 9, 10, 55, 6, 78, 21, 60.
T(14,5) = A109395(14)*5 - A076512(14)*A038566(14,5) = 7*5 - 3*11 = 2.
T(210,2) = A109395(210)*2 - A076512(210)*A038566(210,2) = 35*2 - 8*11 = -18.

Mathematica

Flatten@ 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, 15}] (* Michael De Vlieger, Jun 06 2019 *)

Prog

(PARI) vtot(n) = select(x->(gcd(n, x)==1), vector(n, k, k));
row(n) = my(q = eulerphi(n)/n, v = vtot(n)); vector(#v, k, denominator(q)*k - numerator(q)*v[k]); \\ Michel Marcus, May 14 2019

Crossrefs

Cf. A000010, A109395, A038566, A076511, A076512.

Sequence in context: A035389 A129176 A134132 * A030424 A216656 A353435

Adjacent sequences: A308118 A308119 A308120 * A308122 A308123 A308124

Keyword

sign,tabf

Author

Jamie Morken, May 13 2019

Status

approved