A112543 Numerators of fractions n/k in array by antidiagonals.

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

Offset: 1

Author

Franklin T. Adams-Watters, Sep 11 2005

Keywords

• easy
• frac
• nonn
• tabl

Comments

Column k has period k. - Clark Kimberling, Jul 04 2013

Read as a triangle with antidiagonals for rows, T(n,k) gives the number of distinct locations at which two points the same distance from a center rotating around that center in the same direction at speeds n+1 and k will coincide. - Thomas Anton, Nov 23 2018

As a rectangular array, each row sequence is multiplicative and strongly divisible, as discussed in Comments in A060819. The fractions, as n/gcd(m,n) are generated from an initial generation G(0) = (m/1) by applying the function f(x,y) = x*y/(x+y) = 1/(1/x + 1/y) to successive generations; i.e., for n>=1, G(n) = f(G(n-1)). Duplicates are removed as they occur, leaving 2^m fractions in G(m), for m>=0. See Example. - Clark Kimberling, Aug 04 2025

Examples

			a(2,4) = 1/2 because 2/4 = 1/2.
Northwest corner of the array:
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...;
  1, 1, 3, 2, 5, 3, 7, 4, 9,  5, ...;
  1, 2, 1, 4, 5, 2, 7, 8, 3, 10, ...;
  1, 1, 3, 1, 5, 3, 7, 2, 9,  5, ...;
  1, 2, 3, 4, 1, 6, 7, 8, 9,  2, ...;
  1, 1, 1, 2, 5, 1, 7, 4, 3,  5, ...;
  1, 2, 3, 4, 5, 6, 1, 8, 9, 10, ...;
  1, 1, 3, 1, 5, 3, 7, 1, 9,  5, ...;
  1, 2, 1, 4, 5, 2, 7, 8, 1, 10, ...;
  1, 1, 3, 2, 1, 3, 7, 4, 9,  1, ...;
Antidiagonal triangle begins:
   1;
   2, 1;
   3, 1, 1;
   4, 3, 2, 1;
   5, 2, 1, 1, 1;
   6, 5, 4, 3, 2, 1;
   7, 3, 5, 1, 3, 1, 1;
   8, 7, 2, 5, 4, 1, 2, 1;
   9, 4, 7, 3, 1, 2, 3, 1, 1;
  10, 9, 8, 7, 6, 5, 4, 3, 2, 1;
Regarding the generations G(n) mentioned in Comments, for m = 2, the first 4 generations are G(0) = (2), G(1) = (2,1), G(2) = (2, 1, 2/3, 1/2), G(3) = (2, 1, 2/3, 1/2, 2/5, 1/3, 2/7, 1/4); the denominators in G(3) are (1, 1, 3, 2, 5, 3, 7, 4), as in row 2. - _Clark Kimberling_, Aug 04 2025
		

Links

• Clark Kimberling, Antidiagonals n = 1..60, flattened
• Eric Weisstein's World of Mathematics, Exponential Integral

Crossrefs

Denominators in A112544. Reduced version of A004736/A002260.

Cf. A332049.

Programs

Magma

[Numerator((n-k+1)/k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jan 12 2022

Mathematica

d[m_, n_] := n/GCD[m, n]; z = 12;
TableForm[Table[d[m, n], {m, 1, z}, {n, 1, z}] ] (*array*)
Flatten[Table[d[k, m + 1 - k], {m, 1, z}, {k, 1, m}]] (*sequence*)
(* Clark Kimberling, Jul 04 2013 *)

PARI

t1(n) = binomial(floor(3/2+sqrt(2*n)),2) -n+1;
t2(n) = n - binomial(floor(1/2+sqrt(2*n)),2);
vector(100, n, t1(n)/gcd(t1(n),t2(n)))

Sage

flatten([[numerator((n-k+1)/k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Jan 12 2022

Formula

For n/k: (Start)

G.f.: x/(1-x)*log(1/(1-y)),

E.g.f.: x*exp(x)*(Ei(y) - log(y) + EulerGamma) = x*e^x*Integral_{t=0}^{y} (exp(t) - 1) dt. (End)

T(n, k) = n/gcd(k, n). - Clark Kimberling, Jul 04 2013

From G. C. Greubel, Jan 12 2022: (Start)

A(n, k) = numerator(k/n) (array).

T(n, k) = numerator((n-k+1)/k) (antidiagonals).

Sum_{k=1..n} T(n, k) = A332049(n+1).

T(n, k) = A112544(n, n-k). (End)

Extensions

Keyword tabl added by Franklin T. Adams-Watters, Sep 02 2009