A060753 -id:A060753 - cp's OEIS frontend

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

Michael De Vlieger, Jamie Morken, May 30 2020

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

			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.

Index entries for sequences related to primorial numbers

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

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]}

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).

A335261 Irregular triangle S(n,k) = denominators of k*A002110(n)/A005867(n) for 1 <= k <= A005867(n).

Original entry on oeis.org

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, 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, 16, 2, 16, 8, 16, 4, 16, 8, 16, 1, 16, 8, 16, 4, 16, 8, 16

Offset: 1

Michael De Vlieger, Jamie Morken, May 30 2020

Alternatively, denominators of k*A060753(n)/A038110(n) for 1 <= k <= A005867(n).
Let m = A038110(n). For row n, the primitive denominators d | m.
The mean of row n is related to the mean of row n of A309497: A060753(n+1)/(A038110(n+1)*2) = Mean(A309497(n))/A038110(n+1).

			Table begins:
    1;
    1;
    1, 1;
    4, 2, 4, 1, 4, 2, 4, 1;
    8, 4, 8, 2, 8, 4, 8, 1, ..., 8, 1;
    ...
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.
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.

Cf. A000010, A002110, A005867, A038110, A060753, A309497, A335260.

Table[Denominator[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]; 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]]
(* or, to generate a single denominator of T(n,k) *)
f[n_, k_] := #/GCD[#, Mod[k, #]] &@ Numerator@ Product[1 - 1/Prime[i], {i, n - 1}]

T(n,k) = m/GCD(k (mod m), m) with m = A038110(n).

Row lengths: A005867(n).

cp's OEIS Frontend

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

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