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A326377
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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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1
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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
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OFFSET
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1,2
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COMMENTS
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This sequence is the main diagonal of A326376.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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
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PROG
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(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)))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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