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A240226
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4-adic value of 1/n, n >= 1.
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3
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1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 64, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 64, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4
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OFFSET
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1,2
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COMMENTS
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For the definition of g-adic value of x, called |x|_g with g an integer >= 2, see the Mahler reference, p. 7. Sometimes also called g-adic absolute value of x. If g is not a prime then this is called a non-archimedean pseudo-valuation. See Mahler, p. 10.
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REFERENCES
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Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.
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LINKS
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FORMULA
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a(n) = 1 if n is odd. a(n) = 4^f(1/n) if n is even, where f(1/n) is the smallest positive integer such that the highest power of 2 in n (that is A006519(n)) divides 4^f(1/n). The f(1/n) values are given in A244415(n).
a(n) = 4^valuation(2*n, 4) = 4^A244415(n).
Multiplicative with a(2^e) = 4^ceiling(e/2), a(p^e) = 1 for odd prime p. (End)
Dirichlet g.f.: zeta(s)*(2^s-1)*(2^s+4)/(4^s-4).
Sum_{k=1..n} a(k) ~ (3/(4*log(2))) * n * (log(n) + gamma + 4*log(2)/3 - 1), where gamma is Euler's constant (A001620). (End)
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EXAMPLE
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n = 2: A006519(2) = 1, 2 divides 4^1, hence f(1/2) = 1 and a(2) = 4^1 = 4.
n = 4: A006519(4) = 2^2, 4 divides 4^1, hence f(1/4) = 1 and a(4) = 4.
n = 8: A006519(8) = 2^3, 8 does not divide 4^1 but 4^2, hence f(1/8) = 2 and a(8) = 4^2 = 16.
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MATHEMATICA
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PROG
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(PARI) a(n) = 4^ceil(valuation(n, 2)/2); \\ Andrew Howroyd, Jul 31 2018
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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