Euler's phi function.
if $n$ has a prime factorization $p_1^ip_2^jp_3^k$
$\frac {\phi(n)}{n} = \frac{(p_1-1)(p_2-1)(p_3-1)}{p_1p_2p_3}$
If trying to find the smallest $n$ such than $\phi(n) < \frac 1{10}$ there is no reason for to choose $i,j,k > 1$
Take the smallest consecutive prime numbers until you have enough
$\frac{(2-1)(3-1)(5-1)\cdots}{2\cdot 3\cdot 5\cdots}$
You may need 50 prime numbers, and their product may be beyond what you can compute directly, but will get there eventually.
ADDED: it is known that $$ \phi(n) > \frac{n}{e^\gamma \log \log n + \frac{3}{\log \log n}} $$ for $n>2,$ and where $\gamma = 0.5772156649...$
Once you know to use primorials, you can just add up logs, I used the reciprocal. For each new prime $p,$ I add $$ \log p \; - \; \log (p-1) = \log \left( \frac{p}{p-1} \right) $$ to the cumulative sum
Fri Dec 27 12:30:33 PST 2019
1 Prime 2 log of (primorial) P over P 0.6931471805599453 log ten 2.302585092994046
2 Prime 3 log of (primorial) P over P 1.09861228866811 log ten 2.302585092994046
3 Prime 5 log of (primorial) P over P 1.321755839982319 log ten 2.302585092994046
4 Prime 7 log of (primorial) P over P 1.475906519809577 log ten 2.302585092994046
5 Prime 11 log of (primorial) P over P 1.571216699613903 log ten 2.302585092994046
6 Prime 13 log of (primorial) P over P 1.651259407287438 log ten 2.302585092994046
7 Prime 17 log of (primorial) P over P 1.711884029103873 log ten 2.302585092994046
8 Prime 19 log of (primorial) P over P 1.765951250374148 log ten 2.302585092994046
9 Prime 23 log of (primorial) P over P 1.810403012944982 log ten 2.302585092994046
10 Prime 29 log of (primorial) P over P 1.845494332756253 log ten 2.302585092994046
11 Prime 31 log of (primorial) P over P 1.878284155579243 log ten 2.302585092994046
12 Prime 37 log of (primorial) P over P 1.905683129767358 log ten 2.302585092994046
13 Prime 41 log of (primorial) P over P 1.930375742357729 log ten 2.302585092994046
14 Prime 43 log of (primorial) P over P 1.953906239767923 log ten 2.302585092994046
15 Prime 47 log of (primorial) P over P 1.975412444988887 log ten 2.302585092994046
16 Prime 53 log of (primorial) P over P 1.994460639959582 log ten 2.302585092994046
17 Prime 59 log of (primorial) P over P 2.011555073318883 log ten 2.302585092994046
18 Prime 61 log of (primorial) P over P 2.028084375270094 log ten 2.302585092994046
19 Prime 67 log of (primorial) P over P 2.043122252634634 log ten 2.302585092994046
20 Prime 71 log of (primorial) P over P 2.057306887626591 log ten 2.302585092994046
21 Prime 73 log of (primorial) P over P 2.071100209758927 log ten 2.302585092994046
22 Prime 79 log of (primorial) P over P 2.083839235536357 log ten 2.302585092994046
23 Prime 83 log of (primorial) P over P 2.095960596068702 log ten 2.302585092994046
24 Prime 89 log of (primorial) P over P 2.107260151322635 log ten 2.302585092994046
25 Prime 97 log of (primorial) P over P 2.117622938358182 log ten 2.302585092994046
26 Prime 101 log of (primorial) P over P 2.127573269211351 log ten 2.302585092994046
27 Prime 103 log of (primorial) P over P 2.137329444156716 log ten 2.302585092994046
28 Prime 107 log of (primorial) P over P 2.146719184506555 log ten 2.302585092994046
29 Prime 109 log of (primorial) P over P 2.155935839611478 log ten 2.302585092994046
30 Prime 113 log of (primorial) P over P 2.164824787028725 log ten 2.302585092994046
31 Prime 127 log of (primorial) P over P 2.172729966535838 log ten 2.302585092994046
32 Prime 131 log of (primorial) P over P 2.180392839281407 log ten 2.302585092994046
33 Prime 137 log of (primorial) P over P 2.187718879373481 log ten 2.302585092994046
34 Prime 139 log of (primorial) P over P 2.194939127346967 log ten 2.302585092994046
35 Prime 149 log of (primorial) P over P 2.201673159528312 log ten 2.302585092994046
36 Prime 151 log of (primorial) P over P 2.208317702246981 log ten 2.302585092994046
37 Prime 157 log of (primorial) P over P 2.214707500345751 log ten 2.302585092994046
38 Prime 163 log of (primorial) P over P 2.22086136592013 log ten 2.302585092994046
39 Prime 167 log of (primorial) P over P 2.226867389980343 log ten 2.302585092994046
40 Prime 173 log of (primorial) P over P 2.232664507664669 log ten 2.302585092994046
41 Prime 179 log of (primorial) P over P 2.238266763213339 log ten 2.302585092994046
42 Prime 181 log of (primorial) P over P 2.243806943588955 log ten 2.302585092994046
43 Prime 191 log of (primorial) P over P 2.249056299475098 log ten 2.302585092994046
44 Prime 193 log of (primorial) P over P 2.254251116352203 log ten 2.302585092994046
45 Prime 197 log of (primorial) P over P 2.259340185859674 log ten 2.302585092994046
46 Prime 199 log of (primorial) P over P 2.264377979889631 log ten 2.302585092994046
47 Prime 211 log of (primorial) P over P 2.269128582648229 log ten 2.302585092994046
48 Prime 223 log of (primorial) P over P 2.273622972236069 log ten 2.302585092994046
49 Prime 227 log of (primorial) P over P 2.278037990445186 log ten 2.302585092994046
50 Prime 229 log of (primorial) P over P 2.282414365044985 log ten 2.302585092994046
51 Prime 233 log of (primorial) P over P 2.286715446944375 log ten 2.302585092994046
52 Prime 239 log of (primorial) P over P 2.290908325204412 log ten 2.302585092994046
53 Prime 241 log of (primorial) P over P 2.295066335353075 log ten 2.302585092994046
54 Prime 251 log of (primorial) P over P 2.299058356622613 log ten 2.302585092994046
55 Prime 257 log of (primorial) P over P 2.302956997038271 log ten 2.302585092994046
Fri Dec 27 12:30:33 PST 2019
====================================================
double logten = log(10.0);
cout.precision(16);
cout << logten << endl;
double log_P_over_phi_P = 0.0;
int count = 0;
for(mpz_class n = 2; log_P_over_phi_P < logten && n <= 2000; ++n)
{
if( mp_PrimeQ(n) )
{
++count;
cout << count << " Prime " << n << " ";
log_P_over_phi_P += mp_Log(n);
log_P_over_phi_P -= mp_Log(n-1);
cout << " log of (primorial) P over P " << log_P_over_phi_P << " log ten " << logten << endl;
}
}
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