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

0,6

Links

Table of n, a(n) for n=0..90.

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.

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

Example

  [ 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;

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 *)

Prog

(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)])

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).

Sequence in context: A357180 A196660 A342323 * A135222 A285706 A357181

Adjacent sequences: A374430 A374431 A374432 * A374434 A374435 A374436

Keyword

nonn,tabl,easy

Author

Peter Luschny, Jul 10 2024

Status

approved