A335260 Irregular triangle S(n,k) = numerators of k*A002110(n)/A005867(n) for 1 <= k <= A005867(n).

1, 2, 3, 6, 15, 15, 45, 15, 75, 45, 105, 30, 35, 35, 105, 35, 175, 105, 245, 35, 315, 175, 385, 105, 455, 245, 525, 70, 595, 315, 665, 175, 735, 385, 805, 105, 875, 455, 945, 245, 1015, 525, 1085, 140, 1155, 595, 1225, 315, 1295, 665, 1365, 175, 1435, 735, 1505

Offset: 1

Author

Michael De Vlieger, Jamie Morken, May 30 2020

Keywords

• nonn
• frac
• easy
• tabf

Comments

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

Examples

			Table begins:
     1;
     2;
     3, 6;
    15, 15, 45, 15, 75, 45, 105, 30;
    ...
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.
		

Links

• Index entries for sequences related to primorial numbers

Crossrefs

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

Programs

Mathematica

Table[Numerator[P Range[EulerPhi[P]]/EulerPhi[P]], {P, FoldList[Times, Prime@ Range@ 5]}] (* or, more efficiently for larger datasets: *)
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]]
(* or, generate a single numerator of S(n,k): *)
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]}

Formula

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

Row lengths: A005867(n).

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