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

1,2

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.

Links

Table of n, a(n) for n=1..80.

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).

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|  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

Prog

(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)))

Crossrefs

See A326377 for the main diagonal of T.

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

Sequence in context: A082506 A053000 A002070 * A106052 A050473 A057593

Adjacent sequences: A326373 A326374 A326375 * A326377 A326378 A326379

Keyword

nonn,tabl

Author

Rémy Sigrist, Jul 02 2019

Status

approved