A374436 Triangle read by rows: T(n, k) = Product_{p in PF(n) union PF(k)} p, where PF(a) is the set of the prime factors of a.

1, 1, 1, 2, 2, 2, 3, 3, 6, 3, 2, 2, 2, 6, 2, 5, 5, 10, 15, 10, 5, 6, 6, 6, 6, 6, 30, 6, 7, 7, 14, 21, 14, 35, 42, 7, 2, 2, 2, 6, 2, 10, 6, 14, 2, 3, 3, 6, 3, 6, 15, 6, 21, 6, 3, 10, 10, 10, 30, 10, 10, 30, 70, 10, 30, 10, 11, 11, 22, 33, 22, 55, 66, 77, 22, 33, 110, 11

Offset: 0

Author

Peter Luschny, Jul 10 2024

Keywords

• nonn
• tabl

Examples

			  [ 0]  1;
  [ 1]  1,  1;
  [ 2]  2,  2,  2;
  [ 3]  3,  3,  6,  3;
  [ 4]  2,  2,  2,  6,  2;
  [ 5]  5,  5, 10, 15, 10,  5;
  [ 6]  6,  6,  6,  6,  6, 30,  6;
  [ 7]  7,  7, 14, 21, 14, 35, 42,  7;
  [ 8]  2,  2,  2,  6,  2, 10,  6, 14,  2;
  [ 9]  3,  3,  6,  3,  6, 15,  6, 21,  6,  3;
  [10] 10, 10, 10, 30, 10, 10, 30, 70, 10, 30,  10;
  [11] 11, 11, 22, 33, 22, 55, 66, 77, 22, 33, 110, 11;
		

Crossrefs

Family: A374433 (intersection), A374434 (symmetric difference), A374435 (difference), this sequence (union).

Cf. A007947 (column 0, main diagonal), A099985 (central terms).

Programs

Maple

PF := n -> ifelse(n = 0, {}, NumberTheory:-PrimeFactors(n)):
A374436 := (n, k) -> mul(PF(n) union PF(k)):
seq(print(seq(A374436(n, k), k = 0..n)), n = 0..11);

Mathematica

nn = 12; Do[Set[s[i], FactorInteger[i][[All, 1]]], {i, 0, nn}]; s[0] = {1}; Table[Apply[Times, Union[s[k], s[n]]], {n, 0, nn}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 11 2024 *)

Python

# Function A374436 defined in A374433.
for n in range(12): print([A374436(n, k) for k in range(n + 1)])

Formula

T(0,0) = T(n,0) = 1; T(n,k) = rad(k*n) where rad = A007947. - Michael De Vlieger, Jul 11 2024