1,5

The array T is completely multiplicative in both parameters.

For any n > 0 and prime number p, T(n, p) is the highest power of p dividing n.

For any function f associating a nonnegative value to any pair of nonnegative values and such that f(0, 0) = 0, we can build an analog of this sequence, say P_f, such that for any prime number p and any n and k > 0 with p-adic valuations i and j, the p-adic valuation of P_f(n, k) equals f(i, j):

f(i, j) P_f

------- ---

i * j T (this sequence)

i + j A003991 (product)

abs(i-j) A089913

min(i, j) A003989 (GCD)

max(i, j) A003990 (LCM)

i AND j A059895

i OR j A059896

i XOR j A059897

If log(N) denotes the set {log(n) : n is in N, the set of the positive integers}, one can define a binary operation on log(N): with prime factorizations n = Product p_i^e_i and k = Product p_i^f_i, set log(n) o log(k) = Sum_{i} (e_i*f_i) * log(p_i). o has the premises of a scalar product even if log(N) isn't a vector space. T(n, k) can be viewed as exp(log(n) o log(k)). - Luc Rousseau, Oct 11 2020

T(n, k) = T(k, n) (T is commutative).

T(m, T(n, k)) = T(T(m, n), k) (T is associative).

T(n, k) = 1 iff gcd(n, k) = 1.

T(n, n) = A054496(n).

T(n, A007947(n)) = n.

T(n, 1) = 1.

T(n, 2) = A006519(n).

T(n, 3) = A038500(n).

T(n, 4) = A006519(n)^2.

T(n, 5) = A060904(n).

T(n, 6) = A065331(n).

T(n, 7) = A268354(n).

T(n, 8) = A006519(n)^3.

T(n, 9) = A038500(n)^2.

T(n, 10) = A132741(n).

T(n, 11) = A268357(n).

Array T(n, k) begins:

n\k| 1 2 3 4 5 6 7 8 9 10

---+--------------------------------------------------

1| 1 1 1 1 1 1 1 1 1 1

2| 1 2 1 4 1 2 1 8 1 2 -> A006519

3| 1 1 3 1 1 3 1 1 9 1 -> A038500

4| 1 4 1 16 1 4 1 64 1 4

5| 1 1 1 1 5 1 1 1 1 5 -> A060904

6| 1 2 3 4 1 6 1 8 9 2 -> A065331

7| 1 1 1 1 1 1 7 1 1 1 -> A268354

8| 1 8 1 64 1 8 1 512 1 8

9| 1 1 9 1 1 9 1 1 81 1

10| 1 2 1 4 5 2 1 8 1 10 -> A132741

T[n_, k_] := With[{p = FactorInteger[GCD[n, k]][[All, 1]]}, If[p == {1}, 1, Times @@ (p^(IntegerExponent[n, p] * IntegerExponent[k, p]))]];

Table[T[n-k+1, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 11 2018 *)

(PARI) T(n, k) = my (p=factor(gcd(n, k))[, 1]); prod(i=1, #p, p[i]^(valuation(n, p[i]) * valuation(k, p[i])))

Cf. A003989, A003990, A003991, A006519, A007947, A038500, A054496, A059895, A059896, A059897, A060904, A065331, A089913, A132741, A268354, A268357.

Sequence in context: A036065 A082907 A146532 * A225372 A184879 A373201

Adjacent sequences: A305717 A305718 A305719 * A305721 A305722 A305723

nonn,tabl,mult

Rémy Sigrist, Jun 09 2018

approved