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

%I A112544 #30 Jan 17 2025 19:22:13
%S A112544 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 A112544 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 A112544 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 A112544 Denominators of fractions n/k in array by antidiagonals.
%H A112544 G. C. Greubel, <a href="/A112544/b112544.txt">Antidiagonals n = 1..50, flattened</a>
%F A112544 From _G. C. Greubel_, Jan 12 2022: (Start)
%F A112544 A(n, k) = denominator(n/k) (array).
%F A112544 T(n, k) = denominator((n-k+1)/k) (antidiagonal triangle).
%F A112544 Sum_{k=1..n} T(n, k) = A332049(n+1).
%F A112544 T(n, k) = A112543(n, n-k). (End)
%e A112544 Array begins as:
%e A112544   1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...;
%e A112544   1, 1, 3, 2, 5, 3, 7, 4, 9,  5, ...;
%e A112544   1, 2, 1, 4, 5, 2, 7, 8, 3, 10, ...;
%e A112544   1, 1, 3, 1, 5, 3, 7, 2, 9,  5, ...;
%e A112544   1, 2, 3, 4, 1, 6, 7, 8, 9,  2, ...;
%e A112544   1, 1, 1, 2, 5, 1, 7, 4, 3,  5, ...;
%e A112544   1, 2, 3, 4, 5, 6, 1, 8, 9, 10, ...;
%e A112544   1, 1, 3, 1, 5, 3, 7, 1, 9,  5, ...;
%e A112544   1, 2, 1, 4, 5, 2, 7, 8, 1, 10, ...;
%e A112544   1, 1, 3, 2, 1, 3, 7, 4, 9,  1, ...;
%e A112544 Antidiagonal triangle begins as:
%e A112544   1;
%e A112544   1, 2;
%e A112544   1, 1, 3;
%e A112544   1, 2, 3, 4;
%e A112544   1, 1, 1, 2, 5;
%e A112544   1, 2, 3, 4, 5, 6;
%e A112544   1, 1, 3, 1, 5, 3, 7;
%e A112544   1, 2, 1, 4, 5, 2, 7, 8;
%e A112544   1, 1, 3, 2, 1, 3, 7, 4, 9;
%e A112544   1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
%t A112544 Table[Denominator[(n-k+1)/k], {n,20}, {k,n}]//Flatten (* _G. C. Greubel_, Jan 12 2022 *)
%o A112544 (PARI)
%o A112544 t1(n) = binomial(floor(3/2+sqrt(2*n)),2) -n+1;
%o A112544 t2(n) = n-binomial(floor(1/2+sqrt(2*n)),2);
%o A112544 vector(100,n,t2(n)/gcd(t1(n),t2(n)))
%o A112544 (Magma) [Denominator((n-k+1)/k): k in [1..n], n in [1..20]]; // _G. C. Greubel_, Jan 12 2022
%o A112544 (Sage) flatten([[denominator((n-k+1)/k) for k in (1..n)] for n in (1..20)]) # _G. C. Greubel_, Jan 12 2022
%Y A112544 Numerators in A112543. See comments and references there.
%Y A112544 Cf. A332049.
%K A112544 easy,frac,nonn,tabl
%O A112544 1,3
%A A112544 _Franklin T. Adams-Watters_, Sep 11 2005
%E A112544 Keyword tabl added by _Franklin T. Adams-Watters_, Sep 02 2009