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I would like to write a program in Mathematica to look for the integer $n$ such that the following definition holds for any $1 < m < n $

$$\frac{f(n)}{f(m)}>1+\frac{\log(n/m)}{\log(n) \log(\log(m))}$$

where $f(n)$ is for example number of divisors of $n$ (and also count the number of resulting sets).

I wrote the following code, but the problem is that when it finds the first j1 in the next step of the For, it looks for the value j2 for which

$$\frac{f(j_2)}{f(j_1)}>1+\frac{\log(j_2/j_1)}{\log(j_2) \log(\log( j_1))}$$

but I want it to find the j2 such that

$$\forall j,\ j_1 \le j < j_2 \implies \frac{f(j_2)}{f(j)}>1+\frac{\log(j_2/j)}{\log (j_2) \log(\log(j))}$$

Note: In the following example k[*] is a special and known sub sequence of integers.

i = 20;

count = 0;

list = {i};

For[j = i, j <= 500100, j++,

    If[ f[k[j]]/f[k[i]] > 1 + Log[k[j]/k[i]]/(Log[k[j]] Log[Log[k[i]]]), i = j; 
        AppendTo[list, i];
        count = count + 1
    ]
];

list

Print["count= ", count]

I would like to write a program in Mathematica to look for the integer $n$ such that the following definition holds for any $1 < m < n $

$$\frac{f(n)}{f(m)}>1+\frac{\log(n/m)}{\log(n) \log(\log(m))}$$

where $f(n)$ is for example number of divisors of $n$ (and also count the number of resulting sets).

I wrote the following code, but the problem is that when it finds the first j1 in the next step of the For, it looks for the value j2 for which

$$\frac{f(j_2)}{f(j_1)}>1+\frac{\log(j_2/j_1)}{\log(j_2) \log(\log( j_1))}$$

but I want it to find the j2 such that

$$\forall j,\ j_1 \le j < j_2 \implies \frac{f(j_2)}{f(j)}>1+\frac{\log(j_2/j)}{\log (j_2) \log(\log(j))}$$

Note: In the following example k[*] is a special and known sub sequence of integers.

i = 20;

count = 0;

list = {i};

For[j = i, j <= 500, j++,

    If[ f[k[j]]/f[k[i]] > 1 + Log[k[j]/k[i]]/(Log[k[j]] Log[Log[k[i]]]), i = j; 
        AppendTo[list, i];
        count = count + 1
    ]
];

list

Print["count= ", count]

I would like to write a program in Mathematica to look for the integer $n$ such that the following definition holds for any $1 < m < n $

$$\frac{f(n)}{f(m)}>1+\frac{\log(n/m)}{\log(n) \log(\log(m))}$$

where $f(n)$ is for example number of divisors of $n$ (and also count the number of resulting sets).

I wrote the following code, but the problem is that when it finds the first j1 in the next step of the For, it looks for the value j2 for which

$$\frac{f(j_2)}{f(j_1)}>1+\frac{\log(j_2/j_1)}{\log(j_2) \log(\log( j_1))}$$

but I want it to find the j2 such that

$$\forall j,\ j_1 \le j < j_2 \implies \frac{f(j_2)}{f(j)}>1+\frac{\log(j_2/j)}{\log (j_2) \log(\log(j))}$$

Note: In the following example k[*] is a special and known sub sequence of integers.

i = 20;

count = 0;

list = {i};

For[j = i, j <= 100, j++,

    If[ f[k[j]]/f[k[i]] > 1 + Log[k[j]/k[i]]/(Log[k[j]] Log[Log[k[i]]]), i = j; 
        AppendTo[list, i];
        count = count + 1
    ]
];

list

Print["count= ", count]
6 edited tags
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I would like to write a program in Mathematica to look for the integer $n$ such that the following definition holds for any $1 < m < n $

$$\frac{f(n)}{f(m)}>1+\frac{\log(n/m)}{\log(n) \log(\log(m))}$$

where $f(n)$ is for example number of divisors of $n$ (and also count the number of resulting sets).

I wrote the following code, but the problem is that when it finds the first j1 in the next step of the For, it looks for the value j2 for which

$$\frac{f(j_2)}{f(j_1)}>1+\frac{\log(j_2/j_1)}{\log(j_2) \log(\log( j_1))}$$

but I want it to find the j2 such that

$$\forall j,\ j_1 \le j < j_2 \implies \frac{f(j_2)}{f(j)}>1+\frac{\log(j_2/j)}{\log (j_2) \log(\log(j))}$$

Note: In the following example k[*] is a special and known sub sequence of integers.

i = 20;

count = 0;

list = {i};

For[j = i, j <= 500, j++,

    If[ f[k[j]]/f[k[i]] > 1 + Log[k[j]/k[i]]/(Log[k[j]] Log[Log[k[i]]]), i = j; 
        AppendTo[list, i];
        count = count + 1
    ]
];

list

Print["count= ", count]

I would like to write a program in Mathematica to look for the integer $n$ such that the following definition holds for any $1 < m < n $

$$\frac{f(n)}{f(m)}>1+\frac{\log(n/m)}{\log(n) \log(\log(m))}$$

where $f(n)$ is number of divisors of $n$ (and also count the number of resulting sets).

I wrote the following code, but the problem is that when it finds the first j1 in the next step of the For, it looks for the value j2 for which

$$\frac{f(j_2)}{f(j_1)}>1+\frac{\log(j_2/j_1)}{\log(j_2) \log(\log( j_1))}$$

but I want it to find the j2 such that

$$\forall j,\ j_1 \le j < j_2 \implies \frac{f(j_2)}{f(j)}>1+\frac{\log(j_2/j)}{\log (j_2) \log(\log(j))}$$

Note: In the following example k[*] is a special and known sub sequence of integers.

i = 20;

count = 0;

list = {i};

For[j = i, j <= 500, j++,

    If[ f[k[j]]/f[k[i]] > 1 + Log[k[j]/k[i]]/(Log[k[j]] Log[Log[k[i]]]), i = j; 
        AppendTo[list, i];
        count = count + 1
    ]
];

list

Print["count= ", count]

I would like to write a program in Mathematica to look for the integer $n$ such that the following definition holds for any $1 < m < n $

$$\frac{f(n)}{f(m)}>1+\frac{\log(n/m)}{\log(n) \log(\log(m))}$$

where $f(n)$ is for example number of divisors of $n$ (and also count the number of resulting sets).

I wrote the following code, but the problem is that when it finds the first j1 in the next step of the For, it looks for the value j2 for which

$$\frac{f(j_2)}{f(j_1)}>1+\frac{\log(j_2/j_1)}{\log(j_2) \log(\log( j_1))}$$

but I want it to find the j2 such that

$$\forall j,\ j_1 \le j < j_2 \implies \frac{f(j_2)}{f(j)}>1+\frac{\log(j_2/j)}{\log (j_2) \log(\log(j))}$$

Note: In the following example k[*] is a special and known sub sequence of integers.

i = 20;

count = 0;

list = {i};

For[j = i, j <= 500, j++,

    If[ f[k[j]]/f[k[i]] > 1 + Log[k[j]/k[i]]/(Log[k[j]] Log[Log[k[i]]]), i = j; 
        AppendTo[list, i];
        count = count + 1
    ]
];

list

Print["count= ", count]
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