for each positive integer m, let v(m) denote the number of divisors of m. define the function F(n) =∑ v (d) dIn where the sum is over all positive divisors d of n p...
for each positive integer m, let v(m) denote the number of divisors of m.
define the function F(n) =∑ v (d)
dIn
where the sum is over all positive divisors d of n
prove that function F(n) is multiplicative
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for each positive integer m, let v(m) denote the number of divisors of m. define the function F(n) =∑ v (d) dIn where the sum is over all positive divisors d of n p...
(1) Let f be a multiplicative function satisfying Σ f(d)-n/0(n), where the sum is taken over all ...
(1) Let f be a multiplicative function satisfying Σ f(d)-n/0(n), where the sum is taken over all positive divisors of n, and ф is Euler's totient function. Use the Mobius inversion formula to prove that f(n) ."(n)/0(n)
(1) Let f be a multiplicative function satisfying Σ f(d)-n/0(n), where the sum is taken over all positive divisors of n, and ф is Euler's totient function. Use the Mobius inversion formula to prove that f(n) ."(n)/0(n)
Let f be a multiplicative function satisfying ∑f(d) = n/φ(n)
Let f be a multiplicative function satisfying ∑f(d) = n/φ(n), where the sum is taken over all positive divisors of n, and φ is Euler's totient function. Use the Mobius inversion formula to prove that f(n)=μ2(n)/φ(n)
4) Let D be the set of all finite subsets of positive integers. Define a function...
4) Let D be the set of all finite subsets of positive integers. Define a function (:2 - D as follows: For each positive integer n, f(n) =the set of positive divisors of n. Find the following f (1), f(17) and f(18). Is f one-to-one? Prove or give a counterexample.
T(n) is the number of divisors of n, and u(n)-1 Define an arithmetic function A as...
T(n) is the number of divisors of n, and u(n)-1 Define an arithmetic function A as follows: if p is a prime and k 1 let A(p) log p for all other n, let A(n) 0. (Warning: A is NOT a multiplicative function!) Prove that (A* u)(n) log n for all n. (HINT: consider the various d which divide n expressed in terms of the prime factorization of n
12 ) - 2. Let p(n) denote the number ofdstinct prime divisors ofn. For example, p( p(24)-2 and p(60) 3. Let q(n)an, where a is fixed and show that qn) is multiplicative, but not completely mul...
12 ) - 2. Let p(n) denote the number ofdstinct prime divisors ofn. For example, p( p(24)-2 and p(60) 3. Let q(n)an, where a is fixed and show that qn) is multiplicative, but not completely multiplicative.
12 ) - 2. Let p(n) denote the number ofdstinct prime divisors ofn. For example, p( p(24)-2 and p(60) 3. Let q(n)an, where a is fixed and show that qn) is multiplicative, but not completely multiplicative.
DEFINITION: For a positive integer n, τ(n) is the number of positive divisors of n and...
DEFINITION: For a positive integer n, τ(n) is the number of
positive divisors of n and σ(n) is the sum of those divisors.
4. The goal of this problem is to prove the inequality in part (b), that o(1)+(2)+...+on) < nº for each positive integer n. The first part is a stepping-stone for that. (a) (10 points.) Fix positive integers n and k with 1 <ksn. (i) For which integers i with 1 <i<n is k a term in the...
A perfect number is a positive integer that equals the sum of all of its divisors...
A perfect number is a positive integer that equals the sum of all of its divisors (including the divisor 1 but excluding the number itself). For example 6, 28 and 496 are perfect numbers because 6=1+2+3 28 1 + 2 + 4 + 7 + 14 496 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 Write a program to read a positive integer value, N, and find the smallest perfect number...
I got a C++ problem. Let n be a positive integer and let S(n) denote the...
I got a C++ problem.
Let n be a positive integer and let S(n) denote the number of divisors of n. For example, S(1)- 1, S(4)-3, S(6)-4 A positive integer p is called antiprime if S(n)くS(p) for all positive n 〈P. In other words, an antiprime is a number that has a larger number of divisors than any number smaller than itself. Given a positive integer b, your program should output the largest antiprime that is less than or equal...
Q18 12 Points For any positive integer n, let bn denote the number of n-digit positive...
Q18 12 Points For any positive integer n, let bn denote the number of n-digit positive integers whose digits are all 1 or 2, and have no two consecutive digits of 1. For example, for n - 3, 121 is one such integer, but 211 is not, since it has two consecutive 1 's at the end. Find a recursive formula for the sequence {bn}. You have to fully prove your answer.
1. Let m be a nonnegative integer, and n a positive integer. Using the division algorithm...
1. Let m be a nonnegative integer, and n a positive integer. Using the division algorithm we can write m=qn+r, with 0 <r<n-1. As in class define (m,n) = {mc+ny: I,Y E Z} and S(..r) = {nu+ru: UV E Z}. Prove that (m,n) = S(n,r). (Remark: If we add to the definition of ged that gedan, 0) = god(0, n) = n, then this proves that ged(m, n) = ged(n,r). This result leads to a fast algorithm for computing ged(m,...
(1) Let f be a multiplicative function satisfying Σ f(d)-n/0(n), where the sum is taken over all positive divisors of n, and ф is Euler's totient function. Use the Mobius inversion formula to prove that f(n) ."(n)/0(n)
(1) Let f be a multiplicative function satisfying Σ f(d)-n/0(n), where the sum is taken over all positive divisors of n, and ф is Euler's totient function. Use the Mobius inversion formula to prove that f(n) ."(n)/0(n)
4) Let D be the set of all finite subsets of positive integers. Define a function (:2 - D as follows: For each positive integer n, f(n) =the set of positive divisors of n. Find the following f (1), f(17) and f(18). Is f one-to-one? Prove or give a counterexample.
T(n) is the number of divisors of n, and u(n)-1 Define an arithmetic function A as follows: if p is a prime and k 1 let A(p) log p for all other n, let A(n) 0. (Warning: A is NOT a multiplicative function!) Prove that (A* u)(n) log n for all n. (HINT: consider the various d which divide n expressed in terms of the prime factorization of n
12 ) - 2. Let p(n) denote the number ofdstinct prime divisors ofn. For example, p( p(24)-2 and p(60) 3. Let q(n)an, where a is fixed and show that qn) is multiplicative, but not completely multiplicative.
12 ) - 2. Let p(n) denote the number ofdstinct prime divisors ofn. For example, p( p(24)-2 and p(60) 3. Let q(n)an, where a is fixed and show that qn) is multiplicative, but not completely multiplicative.
DEFINITION: For a positive integer n, τ(n) is the number of
positive divisors of n and σ(n) is the sum of those divisors.
4. The goal of this problem is to prove the inequality in part (b), that o(1)+(2)+...+on) < nº for each positive integer n. The first part is a stepping-stone for that. (a) (10 points.) Fix positive integers n and k with 1 <ksn. (i) For which integers i with 1 <i<n is k a term in the...
A perfect number is a positive integer that equals the sum of all of its divisors (including the divisor 1 but excluding the number itself). For example 6, 28 and 496 are perfect numbers because 6=1+2+3 28 1 + 2 + 4 + 7 + 14 496 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 Write a program to read a positive integer value, N, and find the smallest perfect number...
I got a C++ problem.
Let n be a positive integer and let S(n) denote the number of divisors of n. For example, S(1)- 1, S(4)-3, S(6)-4 A positive integer p is called antiprime if S(n)くS(p) for all positive n 〈P. In other words, an antiprime is a number that has a larger number of divisors than any number smaller than itself. Given a positive integer b, your program should output the largest antiprime that is less than or equal...
Q18 12 Points For any positive integer n, let bn denote the number of n-digit positive integers whose digits are all 1 or 2, and have no two consecutive digits of 1. For example, for n - 3, 121 is one such integer, but 211 is not, since it has two consecutive 1 's at the end. Find a recursive formula for the sequence {bn}. You have to fully prove your answer.
1. Let m be a nonnegative integer, and n a positive integer. Using the division algorithm we can write m=qn+r, with 0 <r<n-1. As in class define (m,n) = {mc+ny: I,Y E Z} and S(..r) = {nu+ru: UV E Z}. Prove that (m,n) = S(n,r). (Remark: If we add to the definition of ged that gedan, 0) = god(0, n) = n, then this proves that ged(m, n) = ged(n,r). This result leads to a fast algorithm for computing ged(m,...
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