A374434 Triangle read by rows: T(n, k) = Product_{p in PF(n) symmetric difference PF(k)} p, where PF(a) is the set of the prime factors of a.

1, 1, 1, 2, 2, 1, 3, 3, 6, 1, 2, 2, 1, 6, 1, 5, 5, 10, 15, 10, 1, 6, 6, 3, 2, 3, 30, 1, 7, 7, 14, 21, 14, 35, 42, 1, 2, 2, 1, 6, 1, 10, 3, 14, 1, 3, 3, 6, 1, 6, 15, 2, 21, 6, 1, 10, 10, 5, 30, 5, 2, 15, 70, 5, 30, 1, 11, 11, 22, 33, 22, 55, 66, 77, 22, 33, 110, 1

Offset

0,4

Links

Table of n, a(n) for n=0..77.

Formula

From Michael De Vlieger, Jul 11 2024: (Start)

T(0,0) = T(n,0) = rad(n)/rad(0) = 1 where rad = A007947;

T(n,k) = rad(k*n)/rad(gcd(k,n))

= A007947(k*n)/A007947(S(n,k)) where S = A050873

= A374436(n,k)/A374433(n,k). (End)

Example

  [ 0]  1;
  [ 1]  1,  1;
  [ 2]  2,  2,  1;
  [ 3]  3,  3,  6,  1;
  [ 4]  2,  2,  1,  6,  1;
  [ 5]  5,  5, 10, 15, 10,  1;
  [ 6]  6,  6,  3,  2,  3, 30,  1;
  [ 7]  7,  7, 14, 21, 14, 35, 42,  1;
  [ 8]  2,  2,  1,  6,  1, 10,  3, 14,  1;
  [ 9]  3,  3,  6,  1,  6, 15,  2, 21,  6,  1;
  [10] 10, 10,  5, 30,  5,  2, 15, 70,  5, 30,   1;
  [11] 11, 11, 22, 33, 22, 55, 66, 77, 22, 33, 110, 1;

Maple

PF := n -> ifelse(n = 0, {}, NumberTheory:-PrimeFactors(n)):
A374434 := (n, k) -> mul(symmdiff(PF(n), PF(k))):
seq(print(seq(A374434(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[Times @@ SymmetricDifference[s[k], s[n]], {n, 0, nn}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 11 2024 *)

Prog

(Python)  # Function A374434 defined in A374433.
for n in range(11): print([A374434(n, k) for k in range(n + 1)])

Crossrefs

Family: A374433 (intersection), this sequence (symmetric difference), A374435 (difference), A374436 (union).

Cf. A007947 (column 0), A000034 (central terms), A050873 (gcd).

Sequence in context: A177762 A109380 A167754 * A372688 A011020 A181512

Adjacent sequences: A374431 A374432 A374433 * A374435 A374436 A374438

Keyword

nonn,tabl

Author

Peter Luschny, Jul 10 2024

Status

approved