|
%I #6 May 31 2020 21:59:47
%S 1,1,1,1,4,2,4,1,4,2,4,1,8,4,8,2,8,4,8,1,8,4,8,2,8,4,8,1,8,4,8,2,8,4,
%T 8,1,8,4,8,2,8,4,8,1,8,4,8,2,8,4,8,1,8,4,8,2,8,4,8,1,16,8,16,4,16,8,
%U 16,2,16,8,16,4,16,8,16,1,16,8,16,4,16,8,16
%N Irregular triangle S(n,k) = denominators of k*A002110(n)/A005867(n) for 1 <= k <= A005867(n).
%C Alternatively, denominators of k*A060753(n)/A038110(n) for 1 <= k <= A005867(n).
%C Let m = A038110(n). For row n, the primitive denominators d | m.
%C The mean of row n is related to the mean of row n of A309497: A060753(n+1)/(A038110(n+1)*2)) = 35/16 = Mean(A309497(n))/A038110(n+1).
%F T(n,k) = m/GCD(k (mod m), m) with m = A038110(n).
%F Row lengths: A005867(n).
%e Table begins:
%e 1;
%e 1;
%e 1, 1;
%e 4, 2, 4, 1, 4, 2, 4, 1;
%e 8, 4, 8, 2, 8, 4, 8, 1, ..., 8, 1;
%e ...
%e Row n = 4 contains the denominators of (35/8)*k for 1 <= k <= A005867(4): 35/8, 35/4, 105/8, 35/2, 175/8, 105/4, 245/8, 35, 315/8, 175/4, 385/8, 105/2, 455/8, 245/4, 525/8, 70, 595/8, 315/4, 665/8, 175/2, 735/8, 385/4, 805/8, 105, 875/8, 455/4, 945/8, 245/2, 1015/8, 525/4, 1085/8, 140, 1155/8, 595/4, 1225/8, 315/2, 1295/8, 665/4, 1365/8, 175, 1435/8, 735/4, 1505/8, 385/2, 1575/8, 805/4, 1645/8, 210.
%e The mean of row 4: A060753(4+1)/(A038110(4+1)*2))* (A005867(4)+1) = (35/(8*2))*(48+1) = (35/16)*49 = 1715/16.
%t Table[Denominator[P Range[EulerPhi[P]]/EulerPhi[P]], {P, FoldList[Times, Prime@ Range@ 5]}] (* or, more efficiently for larger datasets: *)
%t Flatten@ Block[{nn = 7, s, t}, s = Array[Numerator@ Product[1 - 1/Prime[k], {k, # - 1}] &, nn]; t = Nest[Append[#, #[[-1]] (Prime[Length@ #] - 1)]&, {1}, nn]; MapIndexed[Function[{m, D, i}, PadRight[{}, t[[i]], ReplacePart[ConstantArray[0, m], Flatten@ Map[Function[d, Map[# -> d &, m/d Select[Range[d], GCD[#, d] == 1 &]]], D]]]] @@ {#1, Divisors@ #1, First[#2]} &, s]]
%t (* or, to generate a single denominator of T(n,k) *)
%t f[n_, k_] := #/GCD[#, Mod[k, #]] &@ Numerator@ Product[1 - 1/Prime[i], {i, n - 1}]
%Y Cf. A000010, A002110, A005867, A038110, A060753, A309497, A335260.
%K nonn,frac,easy,tabf
%O 1,5
%A _Michael De Vlieger_, _Jamie Morken_, May 30 2020
|
