Results 11 to 20 of about 34 (34)
Completely multiplicative functions arising from simple operations
International Journal of Mathematics and Mathematical Sciences, Volume 2004, Issue 9, Page 431-441, 2004., 2004 Given two multiplicative arithmetic functions, various conditions for their convolution, powers, and logarithms to be completely multiplicative, based on values at the primes, are derived together with their applications.
Vichian Laohakosol, Nittiya Pabhapote
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A rationality condition for the existence of odd perfect numbers
International Journal of Mathematics and Mathematical Sciences, Volume 2003, Issue 20, Page 1261-1293, 2003., 2003 A rationality condition for the existence of odd perfect numbers is used to derive an upper bound for the density of odd integers such that σ(N) could be equal to 2N, where N belongs to a fixed interval with a lower limit greater than 10300. The rationality of the square root expression consisting of a product of repunits multiplied by twice the base ...
Simon Davis
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Some characterizations of specially multiplicative functions
International Journal of Mathematics and Mathematical Sciences, Volume 2003, Issue 37, Page 2335-2344, 2003., 2003 A multiplicative function f is said to be specially multiplicative if there is a completely multiplicative function fA such that f(m)f(n) = ∑d|(m,n)f(mn/d2)fA(d) for all m and n. For example, the divisor functions and Ramanujan′s τ‐function are specially multiplicative functions. Some characterizations of specially multiplicative functions are given in
Pentti Haukkanen
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On the difference of values of the kernel function at consecutive integers
International Journal of Mathematics and Mathematical Sciences, Volume 2003, Issue 67, Page 4249-4262, 2003., 2003 For each positive integer n, set γ(n) = Πp|np. Given a fixed integer k ≠ ±1, we establish that if the ABC‐conjecture holds, then the equation γ(n + 1) − γ(n) = k has only finitely many solutions. In the particular cases k = ±1 , we provide a large family of solutions for each of the corresponding equations.
Jean-Marie De Koninck, Florian Luca
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Characterizing completely multiplicative functions by generalized Möbius functions
International Journal of Mathematics and Mathematical Sciences, Volume 29, Issue 11, Page 633-639, 2002., 2002 Using the generalized Möbius functions, μα, first introduced by Hsu (1995), two characterizations of completely multiplicative functions are given; save a minor condition they read (μαf)−1=μ−αf and fα = μ−αf.
Vichian Laohakosol +2 more
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Sequences and series involving the sequence of composite numbers
International Journal of Mathematics and Mathematical Sciences, Volume 31, Issue 1, Page 31-36, 2002., 2002 Denoting by pn and cn the nth prime number and the nth composite number, respectively, we prove that both the sequence (xn)n≥1, defined by xn=∑k=1n (ck+1−ck) / k−pn / n, and the series ∑n=1∞ (pcn−cpn) / npn are convergent.
Panayiotis Vlamos
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Properties of the function f(x) = x/π(x)
International Journal of Mathematics and Mathematical Sciences, Volume 28, Issue 5, Page 307-311, 2001., 2001 We obtain the asymptotic estimations for ∑k=2nf(k) and ∑k=2n1/f(k), where f(k) = k/π(k), k ≥ 2. We study the expression 2f(x + y) − f(x) − f(y) for integers x, y ≥ 2 and as an application we make several remarks in connection with the conjecture of Hardy and Littlewood.
Panayiotis Vlamos
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Some characterizations of totients
International Journal of Mathematics and Mathematical Sciences, Volume 19, Issue 2, Page 209-217, 1996., 1995 An arithmetical function is said to be a totient if it is the Dirichlet convolution between a completely multiplicative function and the inverse of a completely multiplicative function. Euler′s phi‐function is a famous example of a totient. All completely multiplicative functions are also totients.
Pentti Haukkanen
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The unitary amicable pairs to 108
International Journal of Mathematics and Mathematical Sciences, Volume 18, Issue 2, Page 405-410, 1995., 1994 We present an exhaustive list of the 185 unitary amicable pairs whose smaller number is less than 108 and a new unitary sociable set of four numbers.
Rudolph M. Najar
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An identity for a class of arithmetical functions of several variables
, 1993 International Journal of Mathematics and Mathematical Sciences, Volume 16, Issue 2, Page 355-358, 1993.
Pentti Haukkanen
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