A326377 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; a(n) = g(f(n) o f(n)) (where o denotes function composition).

1, 2, 3, 4, 11, 12, 29, 8, 81, 1100, 59, 48, 101, 195478444, 40425, 16, 157, 648, 229, 440000, 64240097649, 1445390468875226977004, 313, 192, 214358881, 44574662297516497591170630280506162081362246142404, 19683, 9921285858330292941824, 421, 72765000, 547, 32

Offset: 1

Author

Rémy Sigrist, Jul 02 2019

Keywords

• nonn

Comments

This sequence is the main diagonal of A326376.

Examples

			The first terms, alongside the corresponding polynomials, are:
  n   a(n)  f(n)   f(n) o f(n)
  --  ----  -----  -----------
   1     1      0            0
   2     2      1            1
   3     3      x            x
   4     4      2            2
   5    11    x^2          x^4
   6    12    x+1          x+2
   7    29    x^3          x^9
   8     8      3            3
   9    81    2*x          4*x
  10  1100  x^2+1  x^4+2*x^2+2
  11    59    x^4         x^16
  12    48    x+2          x+4
		

Crossrefs

Cf. A060722, A243896, A326376.

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))
a(n) = g(f(n, f(n)))

Formula

a(n) = A326376(n, n).

a(2^k) = 2^k for any k >= 0.

a(3^k) = A060722(k) for any k >= 0.

a(prime(k)) = A243896(k) for any k >= 1 (where prime(k) denotes the k-th prime number).