A326376 Square array T(n, k) read by antidiagonals upwards, n > 0 and k > 0: for any number n > 0, let f(n) be the polynomial of a single indeterminate x where the coefficient of x^e is the prime(1+e)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the polynomials of a single indeterminate x with nonnegative integer coefficients; let g be the inverse of f; T(n, k) = g(f(n) o f(k)) (where o denotes function composition).

1, 2, 1, 1, 2, 1, 4, 2, 2, 1, 1, 4, 3, 2, 1, 2, 2, 4, 4, 2, 1, 1, 4, 5, 4, 5, 2, 1, 8, 2, 6, 16, 4, 6, 2, 1, 1, 8, 7, 8, 11, 4, 7, 2, 1, 2, 4, 8, 256, 10, 90, 4, 8, 2, 1, 1, 4, 9, 8, 17, 12, 17, 4, 9, 2, 1, 4, 2, 10, 16, 8, 47250, 14, 512, 4, 10, 2, 1, 1, 8

Offset: 1

Author

Rémy Sigrist, Jul 02 2019

Keywords

• nonn
• tabl

Comments

This sequence has connections with A297845.

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); as such, f is a homomorphism from the multiplicative group of positive rational numbers to the additive group of polynomials of a single indeterminate x with integer coefficients.

Examples

			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|  2  2   2    2   2      2   2          2        2        2
    3|  1  2   3    4   5      6   7          8        9       10
    4|  4  4   4    4   4      4   4          4        4        4
    5|  1  2   5   16  11     90  17        512      625      550
    6|  2  4   6    8  10     12  14         16       18       20
    7|  1  2   7  256  17  47250  29  134217728  5764801  5656750
    8|  8  8   8    8   8      8   8          8        8        8
    9|  1  4   9   16  25     36  49         64       81      100
   10|  2  4  10   32  22    180  34       1024     1250     1100
The corresponding polynomials are:
  f(n)\f(k)| 0 1 x     2 x^2   x+1             x^3   3  2*x     x^2+1
  ---------+---------------------------------------------------------------------
          0| 0 0 0     0 0     0               0     0  0       0
          1| 1 1 1     1 1     1               1     1  1       1
          x| 0 1 x     2 x^2   x+1             x^3   3  2*x     x^2+1
          2| 2 2 2     2 2     2               2     2  2       2
        x^2| 0 1 x^2   4 x^4   x^2+2*x+1       x^6   9  4*x^2   x^4+2*x^2+1
        x+1| 1 2 x+1   3 x^2+1 x+2             x^3+1 4  2*x+1   x^2+2
        x^3| 0 1 x^3   8 x^6   x^3+3*x^2+3*x+1 x^9   27 8*x^3   x^6+3*x^4+3*x^2+1
          3| 3 3 3     3 3     3               3     3  3       3
        2*x| 0 2 2*x   4 2*x^2 2*x+2           2*x^3 6  4*x     2*x^2+2
      x^2+1| 1 2 x^2+1 5 x^4+1 x^2+2*x+2       x^6+1 10 4*x^2+1 x^4+2*x^2+2
		

Crossrefs

See A326377 for the main diagonal of T.

Cf. A006519, A061142, A297473, A297845, A319525.

Programs

PARI

g(p) = my (c=Vecrev(Vec(p))); prod (i=1, #c, if (c[i], prime(i)^c[i], 1))
f(n, v='x) = my (f=factor(n)); sum (i=1, #f~, f[i,2] * v^(primepi(f[i,1]) - 1))
T(n,k) = g(f(n, f(k)))

Formula

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

- T(1, k) = 1,

- T(2^i, k) = 2^i,

- T(3, k) = k,

- T(3^i, k) = k^i,

- T(5, k) = A297473(k),

- T(6, k) = 2*k,

- T(n, 1) = A006519(n),

- T(n, 2) = A061142(n),

- T(n, 3) = n,

- T(n, 5) = A319525(n),

- T(m*n, k) = T(m, k) * T(n, k).