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A112543 Numerators of fractions n/k in array by antidiagonals.

%I A112543 #33 Aug 11 2025 19:57:59
%S A112543 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 A112543 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 A112543 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 A112543 Numerators of fractions n/k in array by antidiagonals.
%C A112543 Column k has period k. - _Clark Kimberling_, Jul 04 2013
%C A112543 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
%C A112543 As a rectangular array, each row sequence is multiplicative and strongly divisible, as discussed in Comments in A060819. The fractions, as n/gcd(m,n) are generated from an initial generation G(0) = (m/1) by applying the function f(x,y) = x*y/(x+y) = 1/(1/x + 1/y) to successive  generations; i.e., for n>=1, G(n) = f(G(n-1)). Duplicates are removed as they occur, leaving 2^m fractions in G(m), for m>=0. See Example. - _Clark Kimberling_, Aug 04 2025
%H A112543 Clark Kimberling, <a href="/A112543/b112543.txt">Antidiagonals n = 1..60, flattened</a>
%H A112543 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ExponentialIntegral.html">Exponential Integral</a>
%F A112543 For n/k: (Start)
%F A112543 G.f.: x/(1-x)*log(1/(1-y)),
%F A112543 E.g.f.: x*exp(x)*(Ei(y) - log(y) + EulerGamma) = x*e^x*Integral_{t=0}^{y} (exp(t) - 1) dt. (End)
%F A112543 T(n, k) = n/gcd(k, n). - _Clark Kimberling_, Jul 04 2013
%F A112543 From _G. C. Greubel_, Jan 12 2022: (Start)
%F A112543 A(n, k) = numerator(k/n) (array).
%F A112543 T(n, k) = numerator((n-k+1)/k) (antidiagonals).
%F A112543 Sum_{k=1..n} T(n, k) = A332049(n+1).
%F A112543 T(n, k) = A112544(n, n-k). (End)
%e A112543 a(2,4) = 1/2 because 2/4 = 1/2.
%e A112543 Northwest corner of the array:
%e A112543   1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...;
%e A112543   1, 1, 3, 2, 5, 3, 7, 4, 9,  5, ...;
%e A112543   1, 2, 1, 4, 5, 2, 7, 8, 3, 10, ...;
%e A112543   1, 1, 3, 1, 5, 3, 7, 2, 9,  5, ...;
%e A112543   1, 2, 3, 4, 1, 6, 7, 8, 9,  2, ...;
%e A112543   1, 1, 1, 2, 5, 1, 7, 4, 3,  5, ...;
%e A112543   1, 2, 3, 4, 5, 6, 1, 8, 9, 10, ...;
%e A112543   1, 1, 3, 1, 5, 3, 7, 1, 9,  5, ...;
%e A112543   1, 2, 1, 4, 5, 2, 7, 8, 1, 10, ...;
%e A112543   1, 1, 3, 2, 1, 3, 7, 4, 9,  1, ...;
%e A112543 Antidiagonal triangle begins:
%e A112543    1;
%e A112543    2, 1;
%e A112543    3, 1, 1;
%e A112543    4, 3, 2, 1;
%e A112543    5, 2, 1, 1, 1;
%e A112543    6, 5, 4, 3, 2, 1;
%e A112543    7, 3, 5, 1, 3, 1, 1;
%e A112543    8, 7, 2, 5, 4, 1, 2, 1;
%e A112543    9, 4, 7, 3, 1, 2, 3, 1, 1;
%e A112543   10, 9, 8, 7, 6, 5, 4, 3, 2, 1;
%e A112543 Regarding the generations G(n) mentioned in Comments, for m = 2, the first 4 generations are G(0) = (2), G(1) = (2,1), G(2) = (2, 1, 2/3, 1/2), G(3) = (2, 1, 2/3, 1/2, 2/5, 1/3, 2/7, 1/4); the denominators in G(3) are (1, 1, 3, 2, 5, 3, 7, 4), as in row 2. - _Clark Kimberling_, Aug 04 2025
%t A112543 d[m_, n_] := n/GCD[m, n]; z = 12;
%t A112543 TableForm[Table[d[m, n], {m, 1, z}, {n, 1, z}] ] (*array*)
%t A112543 Flatten[Table[d[k, m + 1 - k], {m, 1, z}, {k, 1, m}]] (*sequence*)
%t A112543 (* _Clark Kimberling_, Jul 04 2013 *)
%o A112543 (PARI)
%o A112543 t1(n) = binomial(floor(3/2+sqrt(2*n)),2) -n+1;
%o A112543 t2(n) = n - binomial(floor(1/2+sqrt(2*n)),2);
%o A112543 vector(100, n, t1(n)/gcd(t1(n),t2(n)))
%o A112543 (Magma) [Numerator((n-k+1)/k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Jan 12 2022
%o A112543 (Sage) flatten([[numerator((n-k+1)/k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Jan 12 2022
%Y A112543 Denominators in A112544. Reduced version of A004736/A002260.
%Y A112543 Cf. A332049.
%K A112543 easy,frac,nonn,tabl
%O A112543 1,2
%A A112543 _Franklin T. Adams-Watters_, Sep 11 2005
%E A112543 Keyword tabl added by _Franklin T. Adams-Watters_, Sep 02 2009