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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.
8
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
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
KEYWORD
nonn,tabl,easy
AUTHOR
Peter Luschny, Jul 10 2024
STATUS
approved