A335260 - OEIS

%I #8 May 31 2020 21:59:31

%S 1,2,3,6,15,15,45,15,75,45,105,30,35,35,105,35,175,105,245,35,315,175,

%T 385,105,455,245,525,70,595,315,665,175,735,385,805,105,875,455,945,

%U 245,1015,525,1085,140,1155,595,1225,315,1295,665,1365,175,1435,735,1505

%N Irregular triangle S(n,k) = numerators of k*A002110(n)/A005867(n) for 1 <= k <= A005867(n).

%C Alternatively, numerators of k*A060753(n)/A038110(n) for 1 <= k <= A005867(n).

%H <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>

%F S(n,k) = k*A060753(n)/GCD(k (mod m), m) for m = A038110(n).

%F Row lengths: A005867(n).

%F Least numerator in row n: A060753(n), all numerators are multiples j*A060753(n).

%e Table begins:

%e      1;

%e      2;

%e      3, 6;

%e     15, 15, 45, 15, 75, 45, 105, 30;

%e     ...

%e Row n = 4 contains the numerators 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.

%t Table[Numerator[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]; u = Denominator@ Nest[Append[#, #[[-1]] + (1 - #[[-1]])/Prime[Length@ #]] &, {0}, nn]; MapIndexed[Function[{m, D, i},  u[[i]]*Range[t[[i]]]/ PadRight[{}, t[[i]], ReplacePart[ConstantArray[0, m], Flatten@ Map[Function[d, Map[# -> m/d &, m/d Select[Range[d], GCD[#, d] == 1 &]]], D]]]] @@ {#1, Divisors@ #1, First[#2]} &, s]]

%t (* or, generate a single numerator of S(n,k): *)

%t f[n_, k_] := #2 k/GCD[#1, Mod[k, #1]] & @@ {Numerator@ Product[1 - 1/Prime[i], {i, n - 1}], Denominator@ Last@ Nest[Append[#, #[[-1]] + (1 - #[[-1]])/Prime[Length@ #]] &, {0}, n - 1]}

%Y Cf. A000010, A002110, A005867, A038110, A060753, A309497, A335261.

%K nonn,frac,easy,tabf

%O 1,2

%A _Michael De Vlieger_, _Jamie Morken_, May 30 2020