A112544 Denominators of fractions n/k in array by antidiagonals.

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

Offset: 1

Author

Franklin T. Adams-Watters, Sep 11 2005

Keywords

• easy
• frac
• nonn
• tabl

Examples

			Array begins as:
  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 as:
  1;
  1, 2;
  1, 1, 3;
  1, 2, 3, 4;
  1, 1, 1, 2, 5;
  1, 2, 3, 4, 5, 6;
  1, 1, 3, 1, 5, 3, 7;
  1, 2, 1, 4, 5, 2, 7, 8;
  1, 1, 3, 2, 1, 3, 7, 4, 9;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
		

Links

• G. C. Greubel, Antidiagonals n = 1..50, flattened

Crossrefs

Numerators in A112543. See comments and references there.

Cf. A332049.

Programs

Magma

[Denominator((n-k+1)/k): k in [1..n], n in [1..20]]; // G. C. Greubel, Jan 12 2022

Mathematica

Table[Denominator[(n-k+1)/k], {n,20}, {k,n}]//Flatten (* G. C. Greubel, Jan 12 2022 *)

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,t2(n)/gcd(t1(n),t2(n)))

Sage

flatten([[denominator((n-k+1)/k) for k in (1..n)] for n in (1..20)]) # G. C. Greubel, Jan 12 2022

Formula

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

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

T(n, k) = denominator((n-k+1)/k) (antidiagonal triangle).

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

T(n, k) = A112543(n, n-k). (End)

Extensions

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