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%I #29 Sep 08 2022 08:45:22
%S 1,2,1,3,1,1,4,3,2,1,5,2,1,1,1,6,5,4,3,2,1,7,3,5,1,3,1,1,8,7,2,5,4,1,
%T 2,1,9,4,7,3,1,2,3,1,1,10,9,8,7,6,5,4,3,2,1,11,5,3,2,7,1,5,1,1,1,1,12,
%U 11,10,9,8,7,6,5,4,3,2,1,13,6,11,5,9,4,1,3,5,2,3,1,1,14,13,4,11,2,3,8,7,2,1,4,1,2,1
%N Numerators of fractions n/k in array by antidiagonals.
%C Column k has period k. - _Clark Kimberling_, Jul 04 2013
%C Read as a triangle with antidiagonals for rows, T(n,k) gives the number of distinct locations at which two points the same distance from a center rotating around that center in the same direction at speeds n+1 and k will coincide. - _Thomas Anton_, Nov 23 2018
%H Clark Kimberling, <a href="/A112543/b112543.txt">Antidiagonals n = 1..60, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ExponentialIntegral.html">Exponential Integral</a>
%F For n/k: (Start)
%F G.f.: x/(1-x)*log(1/(1-y)),
%F E.g.f.: x*exp(x)*(Ei(y) - log(y) + EulerGamma) = x*e^x*Integral_{t=0}^{y} (exp(t) - 1) dt. (End)
%F T(n, k) = n/gcd(k, n). - _Clark Kimberling_, Jul 04 2013
%F From _G. C. Greubel_, Jan 12 2022: (Start)
%F A(n, k) = numerator(k/n) (array).
%F T(n, k) = numerator((n-k+1)/k) (antidiagonals).
%F Sum_{k=1..n} T(n, k) = A332049(n+1).
%F T(n, k) = A112544(n, n-k). (End)
%e a(2,4) = 1/2 because 2/4 = 1/2.
%e Northwest corner of the array:
%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:
%e 1;
%e 2, 1;
%e 3, 1, 1;
%e 4, 3, 2, 1;
%e 5, 2, 1, 1, 1;
%e 6, 5, 4, 3, 2, 1;
%e 7, 3, 5, 1, 3, 1, 1;
%e 8, 7, 2, 5, 4, 1, 2, 1;
%e 9, 4, 7, 3, 1, 2, 3, 1, 1;
%e 10, 9, 8, 7, 6, 5, 4, 3, 2, 1;
%t d[m_, n_] := n/GCD[m, n]; z = 12;
%t TableForm[Table[d[m, n], {m, 1, z}, {n, 1, z}] ] (*array*)
%t Flatten[Table[d[k, m + 1 - k], {m, 1, z}, {k, 1, m}]] (*sequence*)
%t (* _Clark Kimberling_, Jul 04 2013 *)
%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, t1(n)/gcd(t1(n),t2(n)))
%o (Magma) [Numerator((n-k+1)/k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Jan 12 2022
%o (Sage) flatten([[numerator((n-k+1)/k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Jan 12 2022
%Y Denominators in A112544. Reduced version of A004736/A002260.
%Y Cf. A332049.
%K easy,frac,nonn,tabl
%O 1,2
%A _Franklin T. Adams-Watters_, Sep 11 2005
%E Keyword tabl added by _Franklin T. Adams-Watters_, Sep 02 2009
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