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A112544
Denominators of fractions n/k in array by antidiagonals.
2
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
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
KEYWORD
easy,frac,nonn,tabl
AUTHOR
EXTENSIONS
Keyword tabl added by Franklin T. Adams-Watters, Sep 02 2009
STATUS
approved