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On quotients of values of Euler’s function on the Catalan numbers. (English) Zbl 1409.11004

Summary: In a recent work, F. Luca and P. Stănică [J. Number Theory 132, No. 7, 1404–1424 (2012; Zbl 1271.11005)] examined quotients of the form \(\frac{\varphi(C_m)}{\varphi(C_n)}\), where \(\varphi\) is Euler’s totient function and \(C_0, C_1, C_2 \ldots\) is the sequence of the Catalan numbers. They observed that the number 4 (and analogously \(\frac{1}{4}\)) appears noticeably often as a value of these quotients. We give an explanation of this phenomenon, based on Dickson’s conjecture. It turns out not only that the value 4 is (in a certain sense) special in relation to the quotients \(\frac{\varphi(C_{n + 1})}{\varphi(C_n)}\), but also that the value \(4^k\) has similar “special” properties with respect to the quotients \(\frac{\varphi(C_{n + k})}{\varphi(C_n)}\), and in particular we show that Dickson’s conjecture implies that, for each \(k\), the number \(4^k\) appears infinitely often as a value of the quotients \(\frac{\varphi(C_{n + k})}{\varphi(C_n)}\).

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
11N37 Asymptotic results on arithmetic functions
11B65 Binomial coefficients; factorials; \(q\)-identities
11A63 Radix representation; digital problems

Citations:

Zbl 1271.11005
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Full Text: DOI

References:

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