A374433 Triangle read by rows: T(n, k) = Product_{p in PF(n) intersect PF(k)} p, where PF(a) is the set of the prime factors of a.

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

Offset: 0

Author

Peter Luschny, Jul 10 2024

Keywords

• nonn
• tabl
• easy

Examples

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

Links

• Michael De Vlieger, Plot T(n,k) at (x,y) = (k,-n), n = 0..1024, showing 1 in gray, primes in red, and composites in green.

Crossrefs

Family: this sequence (intersection), A374434 (symmetric difference), A374435 (difference), A374436 (union).

Cf. A007947 (main diagonal and central terms), A374432 (row sums), A374431 (row product).

Programs

Maple

PF := n -> ifelse(n = 0, {}, NumberTheory:-PrimeFactors(n)):
A374433 := (n, k) -> mul(PF(n) intersect PF(k)):
seq(seq(A374433(n, k), k = 0..n), n = 0..12);

Mathematica

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

Python

from math import prod
from sympy import primefactors
def PF(n): return set(primefactors(n)) if n > 0 else set({})
def PrimeIntersect(n, k): return prod(PF(n).intersection(PF(k)))
def PrimeSymDiff(n, k): return prod(PF(n).symmetric_difference(PF(k)))
def PrimeUnion(n, k): return prod(PF(n).union(PF(k)))
def PrimeDiff(n, k): return prod(PF(n).difference(PF(k)))
A374433 = PrimeIntersect; A374434 = PrimeSymDiff
A374435 = PrimeDiff; A374436 = PrimeUnion
for n in range(11): print([A374433(n, k) for k in range(n + 1)])

Formula

T(n, k) = 1 for k = 0, for k > 0: T(n, k) = rad(gcd(n, k)), where rad = A007947 and gcd = A050873. - Michael De Vlieger, Jul 11 2024