A374435 Triangle read by rows: T(n, k) = Product_{p in PF(n) difference PF(k)} p, where PF(a) is the set of the prime factors of a.

1, 1, 1, 2, 2, 1, 3, 3, 3, 1, 2, 2, 1, 2, 1, 5, 5, 5, 5, 5, 1, 6, 6, 3, 2, 3, 6, 1, 7, 7, 7, 7, 7, 7, 7, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 3, 3, 1, 3, 3, 1, 3, 3, 1, 10, 10, 5, 10, 5, 2, 5, 10, 5, 10, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1

Offset: 0

Author

Peter Luschny, Jul 10 2024

Keywords

• nonn
• tabl

Examples

			  [ 0]  1;
  [ 1]  1,  1;
  [ 2]  2,  2,  1;
  [ 3]  3,  3,  3,  1;
  [ 4]  2,  2,  1,  2,  1;
  [ 5]  5,  5,  5,  5,  5,  1;
  [ 6]  6,  6,  3,  2,  3,  6,  1;
  [ 7]  7,  7,  7,  7,  7,  7,  7,  1;
  [ 8]  2,  2,  1,  2,  1,  2,  1,  2,  1;
  [ 9]  3,  3,  3,  1,  3,  3,  1,  3,  3,  1;
  [10] 10, 10,  5, 10,  5,  2,  5, 10,  5, 10,  1;
  [11] 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1;
		

Crossrefs

Family: A374433 (intersection), A374434 (symmetric difference), this sequence (difference), A374436 (union).

Cf. A007947 (column 0), A000034 (central terms).

Programs

Maple

PF := n -> ifelse(n = 0, {}, NumberTheory:-PrimeFactors(n)):
A374435 := (n, k) -> mul(PF(n) minus PF(k)):
seq(print(seq(A374435(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, Complement[s[n], s[k]]], {n, 0, nn}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 11 2024 *)

Python

# Function A374435 defined in A374433.
for n in range(12): print([A374435(n, k) for k in range(n + 1)])