NOTES ON A CURIOUS ARITHMETIC FUNCTION
2011 ◽
Vol 04
(04)
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pp. 705-714
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Keyword(s):
Let v2(n) denote the 2-adic valuation of any positive integer n. Recently, Farhi introduced a curious arithmetic function f defined for any positive integer n by [Formula: see text]. Farhi showed that the inequality [Formula: see text] with c = 4.01055487… holds for all positive integer n and conjectured that one can replace the upper bound cn by 4n in this inequality. In this paper, we show two identities about the product [Formula: see text] and then use it to prove partially Farhi's conjecture. Finally, we propose a conjecture from which the truth of Farhi's conjecture can be deduced. In particular, we confirm the truth of our conjecture for all positive integers n up to 100000 by using Matlab 7.1.
Keyword(s):
2011 ◽
Vol 54
(2)
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pp. 431-441
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1977 ◽
Vol 20
(2)
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pp. 249-254
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