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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).
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%I #10 Jul 06 2019 02:20:51

%S 1,2,3,4,11,12,29,8,81,1100,59,48,101,195478444,40425,16,157,648,229,

%T 440000,64240097649,1445390468875226977004,313,192,214358881,

%U 44574662297516497591170630280506162081362246142404,19683,9921285858330292941824,421,72765000,547,32

%N 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).

%C This sequence is the main diagonal of A326376.

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

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

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

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

%e The first terms, alongside the corresponding polynomials, are:

%e n a(n) f(n) f(n) o f(n)

%e -- ---- ----- -----------

%e 1 1 0 0

%e 2 2 1 1

%e 3 3 x x

%e 4 4 2 2

%e 5 11 x^2 x^4

%e 6 12 x+1 x+2

%e 7 29 x^3 x^9

%e 8 8 3 3

%e 9 81 2*x 4*x

%e 10 1100 x^2+1 x^4+2*x^2+2

%e 11 59 x^4 x^16

%e 12 48 x+2 x+4

%o (PARI) g(p) = my (c=Vecrev(Vec(p))); prod (i=1, #c, if (c[i], prime(i)^c[i], 1))

%o f(n, v='x) = my (f=factor(n)); sum (i=1, #f~, f[i, 2] * v^(primepi(f[i, 1]) - 1))

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

%Y Cf. A060722, A243896, A326376.

%K nonn

%O 1,2

%A _Rémy Sigrist_, Jul 02 2019