A324592 Square array T(n, k) read by diagonals, n > 0, k > 0; for any number m > 0 with prime factorization Product_{i > 0} prime(i)^e(i), let f(m) = Sum_{i > 0} e(i) * sqrt(A005117(i)); f establishes a bijection between the positive numbers and the finite sums of square roots of squarefree numbers; let g be the inverse of f; T(n, k) = g(f(n) * f(k)).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 4, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 11, 16, 11, 6, 1, 1, 7, 12, 25, 25, 12, 7, 1, 1, 8, 17, 36, 8, 36, 17, 8, 1, 1, 9, 27, 49, 55, 55, 49, 27, 9, 1, 1, 10, 16, 64, 31, 72, 31, 64, 16, 10, 1, 1, 11, 33, 81, 125, 119, 119, 125

Offset

1,5

Comments

The set of square roots of squarefree numbers, { sqrt(A005117(i)), i > 0 }, is Q-linearly independent. The set of finite sums of square roots of squarefree numbers is closed under multiplication, hence the sequence is well defined.

The function f can be naturally extended to the set of positive rational numbers: if r = u/v (not necessarily in reduced form), then f(r) = f(u) - f(v).

This sequence has similarities with A297845.

Links

Table of n, a(n) for n=1..74.

Eric Jaffe, Linearly Independent Integer Roots over the Scalar Field Q

Rémy Sigrist, PARI program for A324592

Formula

For any m > 0, n > 0 and k > 0:

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

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

- T(m, n*k) = T(m, n) * T(m, k) and T(n*k, m) = T(n, m) * T(k, m) (T is completely multiplicative in both parameters),

- T(n, 1) = 1 (1 is an absorbing element for T),

- T(n, 2) = n (2 is an identity element for T),

- T(n, 2^i) = n^i for any i >= 0,

- A001221(T(n, k)) <= A001221(n) * A001221(k),

- T(prime(n), prime(n)) = 2^A005117(n) (where prime(n) denotes the n-th prime number).

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   3    4    5    6    7     8     9    10
    3|  1   3   4    9   11   12   17    27    16    33
    4|  1   4   9   16   25   36   49    64    81   100
    5|  1   5  11   25    8   55   31   125   121    40
    6|  1   6  12   36   55   72  119   216   144   330
    7|  1   7  17   49   31  119   32   343   289   217
    8|  1   8  27   64  125  216  343   512   729  1000
    9|  1   9  16   81  121  144  289   729   256  1089
   10|  1  10  33  100   40  330  217  1000  1089   400
For n = 3 and k = 5:
- f(3) = f(prime(2)) = sqrt(A005117(2)) = sqrt(2),
- f(5) = f(prime(3)) = sqrt(A005117(3)) = sqrt(3),
- f(3) * f(5) = sqrt(6) = sqrt(A005117(5)),
- hence T(3, 5) = prime(5) = 11.

Prog

(PARI) See Links section.

Crossrefs

Cf. A001221, A005117, A297845.

Sequence in context: A049695 A096589 A176427 * A099573 A107430 A330885

Adjacent sequences: A324589 A324590 A324591 * A324593 A324594 A324595

Keyword

nonn,tabl

Author

Rémy Sigrist, Sep 03 2019

Status

approved