A305720 Square array T(n, k) read by antidiagonals, n > 0 and k > 0; for any prime number p, the p-adic valuation of T(n, k) is the product of the p-adic valuations of n and of k.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 16, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 8, 1, 4, 5, 4, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 9, 64, 1, 6, 1, 64, 9, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 4, 1, 8, 7, 8

Offset

1,5

Comments

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

Links

Table of n, a(n) for n=1..86.

Formula

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).

Example

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

Mathematica

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 *)

Prog

(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])))

Crossrefs

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 A119335

Adjacent sequences: A305717 A305718 A305719 * A305721 A305722 A305723

Keyword

nonn,tabl,mult

Author

Rémy Sigrist, Jun 09 2018

Status

approved