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
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
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