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

1,3

Links

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

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)

Example

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;

Mathematica

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

Prog

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

Crossrefs

Numerators in A112543. See comments and references there.

Cf. A332049.

Sequence in context: A188542 A029284 A123275 * A198789 A134521 A131375

Adjacent sequences: A112541 A112542 A112543 * A112545 A112546 A112547

Keyword

easy,frac,nonn,tabl

Author

Franklin T. Adams-Watters, Sep 11 2005

Extensions

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

Status

approved