|
%I #26 Sep 08 2022 08:45:22
%S 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,
%T 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,
%U 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
%N Denominators of fractions n/k in array by antidiagonals.
%H G. C. Greubel, <a href="/A112544/b112544.txt">Antidiagonals n = 1..50, flattened</a>
%F From _G. C. Greubel_, Jan 12 2022: (Start)
%F A(n, k) = denominator(n/k) (array).
%F T(n, k) = denominator((n-k+1)/k) (antidiagonal triangle).
%F Sum_{k=1..n} T(n, k) = A332049(n+1).
%F T(n, k) = A112543(n, n-k). (End)
%e Array begins as:
%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...;
%e 1, 1, 3, 2, 5, 3, 7, 4, 9, 5, ...;
%e 1, 2, 1, 4, 5, 2, 7, 8, 3, 10, ...;
%e 1, 1, 3, 1, 5, 3, 7, 2, 9, 5, ...;
%e 1, 2, 3, 4, 1, 6, 7, 8, 9, 2, ...;
%e 1, 1, 1, 2, 5, 1, 7, 4, 3, 5, ...;
%e 1, 2, 3, 4, 5, 6, 1, 8, 9, 10, ...;
%e 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, ...;
%e 1, 2, 1, 4, 5, 2, 7, 8, 1, 10, ...;
%e 1, 1, 3, 2, 1, 3, 7, 4, 9, 1, ...;
%e Antidiagonal triangle begins as:
%e 1;
%e 1, 2;
%e 1, 1, 3;
%e 1, 2, 3, 4;
%e 1, 1, 1, 2, 5;
%e 1, 2, 3, 4, 5, 6;
%e 1, 1, 3, 1, 5, 3, 7;
%e 1, 2, 1, 4, 5, 2, 7, 8;
%e 1, 1, 3, 2, 1, 3, 7, 4, 9;
%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
%t Table[Denominator[(n-k+1)/k], {n,20}, {k,n}]//Flatten (* _G. C. Greubel_, Jan 12 2022 *)
%o (PARI)
%o t1(n) = binomial(floor(3/2+sqrt(2*n)),2) -n+1;
%o t2(n) = n-binomial(floor(1/2+sqrt(2*n)),2);
%o vector(100,n,t2(n)/gcd(t1(n),t2(n)))
%o (Magma) [Denominator((n-k+1)/k): k in [1..n], n in [1..20]]; // _G. C. Greubel_, Jan 12 2022
%o (Sage) flatten([denominator((n-k+1)/k) for k in (1..n)] for n in (1..20)]) # _G. C. Greubel_, Jan 12 2022
%Y Numerators in A112543. See comments and references there.
%Y Cf. A332049.
%K easy,frac,nonn,tabl
%O 1,3
%A _Franklin T. Adams-Watters_, Sep 11 2005
%E Keyword tabl added by _Franklin T. Adams-Watters_, Sep 02 2009
|
