Newsgroup: it.hobby.enigmi
Titolo: I quattro "4"

0 = 4-4+4-4
1 = 4/4x4/4
2 = 4/4+4/4
3 = 4-4^(4-4)
4 = 4+(4-4)x4
5 = 4+4^(4-4)
6 = 4+(4+4)/4
7 = 4+4-4/4
8 = 4+4+4-4
9 = 4+4+4/4
10 = (44-4)/4
11 = 44/sqr(4)/sqr(4)
12 = 4x(4-4/4)
13 = 44/4+sqr(4)
14 = 4+4+4+sqr(4)
15 = 44/4+4
16 = 4x4+4-4
17 = 4x4+4/4
18 = 4x4+4-sqr(4)
19 = 4!-4-4/4
20 = 4x4+sqr(4)+sqr(4)
21 = 4!-4+4/4
22 = 4!-sqr(4)+4-4
23 = 4!-4^(4-4)
24 = 4x4+4+4
25 = 4!+4^(4-4)
26 = 4!+sqr(4)+4-4
27 = 4!+4/4+sqr(4)
28 = 4!+4+4-4
29 = 4!+4+(4/4)=29
30 = 4!+(4-(4/4))!
31 = sqr(4)^int(ln(4)+4)-int(ln(4))=31
32 = 4x4x4/sqr(4)
33 = (4^int(ln(4!))+sqr(4))/sqr(4)
34 = 4!+4+4+sqr(4)
35 = 4x4x sqr(4)+int(ln(4!))
36 = 44-4-4
37 = 44-4-int(ln(4!))
38 = 44-4-sqr(4)
39 = 44-4-int(ln(4))
40 = 44-sqr(4)-sqr(4)
41 = 44-sqr(4)-int(ln(4))
42 = 44-4/sqr(4)
43 = 44-4+int(ln(4!))
44 = 44+4-4
45 = 44+4/4
46 = 44+4-sqr(4)
47 = 44+4-int(ln(4))
48 = 44+sqr(4)+sqr(4)
49 = 44+4+int(ln(4))
50 = 44+4+sqr(4)
51 = 44+4+int(ln(4!))
52 = 44+4+4
53 = 44+(int(ln(4!)))^sqr(4)
54 = (4+sqr(4)) x (int(ln(4!)))^sqr(4)
55 = 4^int(ln(4!)) - (int(ln(4!)))^sqr(4)
56 = 4^int(ln(4!))-4-4
57 = 4^int(ln(4!))-4-int(ln(4!))
58 = 4^int(ln(4!))-4-(4)
59 = 4^int(ln(4!))-4-int(ln(4))
60 = 4x4x4-4
61 = 4x4x4-int(ln(4!))
62 = 4x4x4-sqr(4)
63 = 4x4x4-int(ln(4))
64 = 4^int(ln(4!))+4-4
65 = 4^int(ln(4!))+4/4
66 = 4x4x4+sqr(4)
67 = 4x4x4+int(ln(4!))
68 = 4x4x4+4
69 = 4^int(ln(4!))+4+int(ln(4))
70 = 4^int(ln(4!))+4+sqr(4)
71 = 4^int(ln(4!))+4+int(ln(4!))
72 = 4^int(ln(4!))+4+4
73 = 4^int(ln(4!))+(int(ln(4!)))^sqr(4)
74 = 4^int(ln(4!))+4+(int(ln(4!)))!
75 = (int(ln(4!)))!x4x int(ln(4!)) +int(ln(4!))
76 = 4^int(ln(4!)) + (int(ln(4!)))! x sqr(4)
77 = 4! x int(ln(4!)) +4+int(ln(4))
78 = 4! x int(ln(4!)) +4+sqr(4)
79 = 4! x int(ln(4!)) +4+int(ln(4!))
80 = 4! x int(ln(4!)) +4+4
81 = 4! x int(ln(4!)) +(int(ln(4!)))^sqr(4)
82 = 44 x sqr(4) - (int(ln(4!)))!
83 = (int(ln(4!)))^(sqr(4) x sqr(4)) + sqr(4)
84 = (int(ln(4!)))! x (4+int(ln(4!))) x sqr(4)
85 = (int(ln(4!)))^(sqr(4) x sqr(4)) + 4
86 = (int(ln(4!)))^4+4+int(ln(4))
87 = (int(ln(4!)))^4+4+sqr(4)
88 = 44x4/sqr(4)
89 = 44xsqr(4)+int(ln(4))
90 = 44xsqr(4)+sqr(4)
91 = 44xsqr(4)+int(ln(41))
92 = 44xsqr(4)+4
93 = 4x4!-sqr(4)-int(ln(4))
94 = 4x4!-4+sqr(4)
95 = 4x4!-4+int(ln(4!))
96 = 4x4!+4-4
97 = 4x4!+sqr(4)-int(ln(4))
98 = 4x4!+4/sqr(4)
99 = 4x4!+4-int(ln(4))
100 = 4x4!+sqr(4)+sqr(4)
 
Quale sara' il primo intero non rappresentabile da quattro 4?
 
LZ
 
(io non lo so)
 

 (io non lo so)
 
Ciao!
 
-- 
Yoda
 


444+int(ln(4))
 
:-)
Ciao!

 :-)
 
Ho smirato di poco :-(  ma vediamo adesso come lo fai il 3: 447 ciao
 
Apropos. Ma il .4 americano vale? perche' int(ln .4)=0, altrimenti anche
444 mi sa che non lo becchi ariciao
 
-- 
Yoda
 


444 = 444 x int(ln(4))
447 = 444 + int(ln(4!))
 
:-)

 :-)
 
Ok con tre cifre sei un drago, provo quest'ultimo tentativo 4440, anzi
4447 che e' primo, ma mi sa che verro' battuto da un numero minore ciao
 
-- 
Yoda
 


Per 4447 ho bisogno di sei "4":
4447 = ((int(ln(4!)))!)! x (int(ln(4!)))! + 4^4/sqr(4) - int(ln(4))
Ma penso che il primo intero non fattibile con quattro "4" sia molto piu' piccolo.
 
Ciao!


Non e' vero, bastavano cinque :-)

 Non e' vero, bastavano cinque :-)
 
Occhio anche al semifattoriale (non dico per questo caso) che ad esempio
in un colpo solo ti da' 8 = 4!! ciao
 
-- 
Yoda
 


Ottimo suggerimento, infatti mi ero impantanato su 197:
 
196 = (4!+4!)x4 + 4
197 = (4!!)!!/sqr(4)+4+int(ln(4))
198 = (4^int(ln(4!))+sqr(4)) x int(ln(4!))
199 = (4!!)!!/sqr(4)+4+int(ln(4!))
200 = (4!+4!+sqr(4)) x 4
201 = (4!!)!!/sqr(4) + (int(ln(4!)))^sqr(4)
 

 ...

101 = 4x4!+4+int(ln(4))
102 = 4x4!+4+sqr(4)
103 = 4x4!+4+int(ln(4!))
104 = 4x4!+4+4
105 = 4x4!+(int(ln(4!)))^sqr(4)
106 = (int(ln(4!)))^(int(ln(4!)))x4-sqr(4)
107 = (int(ln(4!)))^(int(ln(4!)))x4-int(ln(4))
108 = (4!+4)x4-4
109 = (4!+4)x4-int(ln(4!))
110 = (4!+4)x4-sqr(4)
111 = (4!+4)x4-int(ln(4))
112 = (4!+4)x4 x int(ln(4))
113 = (4!+4)x4+int(ln(4))
114 = (4!+4)x4+sqr(4)
115 = (4!+4)x4+int(ln(4!))
116 = (4!+4)x4+4
117 = (4!+(int(ln(4!)))!)x4-int(ln(4!))
118 = (4!+(int(ln(4!)))!)x4-sqr(4)
119 = (4!+(int(ln(4!)))!)x4-int(ln(4))
120 = (4!+(int(ln(4!)))!)x4 x int(ln(4))
121 = (4!+(int(ln(4!)))!)x4+int(ln(4))
122 = (4!+(int(ln(4!)))!)x4+sqr(4)
123 = (4!+(int(ln(4!)))!)x4+int(ln(4!))
124 = 4^4/sqr(4)-4
125 = 4^4/sqr(4)-int(ln(4!))
126 = 4^4/sqr(4)-sqr(4)
127 = 4^4/sqr(4)-int(ln(4))
128 = 4^4/sqr(4) x int(ln(4))
129 = 4^4/sqr(4)+int(ln(4)
130 = 4^4/sqr(4)+sqr(4)
131 = 4^4/sqr(4)+int(ln(4!))
132 = 4^4/sqr(4)+4
133 = 44 x Int(ln(4!)) + int(ln(4))
134 = 44 x Int(ln(4!)) + sqr(4)
135 = 44 x Int(ln(4!)) + int(ln(4!))
136 = 44 x Int(ln(4!)) + 4
137 = 4!x(int(ln(4!)))! -4-int(ln(4!))
138 = 4!x(int(ln(4!)))! -4-sqr(4)
139 = 4!x(int(ln(4!)))! -4-int(ln(4))
140 = 4!x(int(ln(4!)))! -sqr(4)-sqr(4)
141 = 4!x(int(ln(4!)))! - int(ln(4!)) x int(ln(4))
142 = 4!x(int(ln(4!)))! - sqr(4) x int(ln(4))
143 = 4!x(int(ln(4!)))! - int(ln(4)) x int(ln(4))
144 = 4!x(int(ln(4!)))! +4-4
145 = 4!x(int(ln(4!)))! +4-int(ln(4!))
146 = 4!x(int(ln(4!)))! +4-sqr(4)
147 = 4!x(int(ln(4!)))! +4-int(ln(4))
148 = 4!x(int(ln(4!)))! +4 x int(ln(4))
149 = 4!x(int(ln(4!)))! +4 + int(ln(4))
150 = 4!x(int(ln(4!)))! +4+sqr(4)
151 = 4!x(int(ln(4!)))! +4+int(ln(4!))
152 = 4!x(int(ln(4!)))! +4+4
153 = (4!+sqr(4)) x (int(ln(4!)))! - int(ln(4!))
154 = (4!+sqr(4)) x (int(ln(4!)))! - sqr(4)
155 = (4!+sqr(4)) x (int(ln(4!)))! - int(ln(4))
156 = (4!+sqr(4)) x (int(ln(4!)))! x int(ln(4))
157 = (4!+sqr(4)) x (int(ln(4!)))! + int(ln(4))
158 = (4!+sqr(4)) x (int(ln(4!)))! + sqr(4)
159 = (4!+sqr(4)) x (int(ln(4!)))! + int(ln(4!))
160 = (44-4)x4
161 = (4!+int(ln(4!))) x (int(ln(4!)))! - int(ln(4))
162 = (4!+int(ln(4!))) x (int(ln(4!)))! x int(ln(4))
163 = (4!+int(ln(4!))) x (int(ln(4!)))! + int(ln(4))
164 = (4!+int(ln(4!))) x (int(ln(4!)))! + sqr(4)
165 = (4!+int(ln(4!))) x (int(ln(4!)))! + int(ln(4!))
166 = (4!+int(ln(4!))) x (int(ln(4!)))! + 4
167 = (4!+4) x (int(ln(4!)))! - int(ln(4))
168 = (4!+4) x (int(ln(4!)))! x int(ln(4))
169 = (4!+4) x (int(ln(4!)))! + int(ln(4))
170 = (4!+4) x (int(ln(4!)))! + sqr(4)
171 = (4!+4) x (int(ln(4!)))! + int(ln(4!))
172 = 44x4 - 4
173 = 44x4 - int(ln(4!))
174 = 44x4 - sqr(4)
175 = 44x4 - int(ln(4))
176 = 44x4 x int(ln(4))
177 = 44x4 + int(ln(4))
178 = 44x4 + sqr(4)
179 = 44x4 + int(ln(4!))
180 = 44x4 + 4
181 = (4!+(int(ln(4!)))!) x int(ln(4!)))! + int(ln(4))
182 = (4!+(int(ln(4!)))!) x int(ln(4!)))! + sqr(4)
183 = (4!+(int(ln(4!)))!) x int(ln(4!)))! + int(ln(4!))
184 = (4!+(int(ln(4!)))!) x int(ln(4!)))! + 4
185 = ((int(ln(4!)))!)!/4 +4+int(ln(4))
186 = ((int(ln(4!)))!)!/4 +4+sqr(4)
187 = ((int(ln(4!)))!)!/4 +4+int(ln(4!))
188 = ((int(ln(4!)))!)!/4 +4+4
189 = ((int(ln(4!)))!)!/4 +(int(ln(4!)))^sqr(4)
190 = (4!+4!)x4 - sqr(4)
191 = (4!+4!)x4 - int(ln(4))
192 = (44+4)x4
193 = (4!+4!)x4 + int(ln(4))
194 = (4!+4!)x4 + sqr(4)
195 = (4!+4!)x4 + int(ln(4!))
196 = (4!+4!)x4 + 4
197 = (4!!)!!/sqr(4)+4+int(ln(4))
198 = (4^int(ln(4!))+sqr(4)) x int(ln(4!))
199 = (4!!)!!/sqr(4)+4+int(ln(4!))
200 = (4!+4!+sqr(4)) x 4
201 = (4!!)!!/sqr(4) + (int(ln(4!)))^sqr(4)
202 = ((4!!)!!+4!)/sqr(4)-sqr(4)
203 = ((4!!)!!+4!)/sqr(4)-int(ln(4))
204 = ((4!!)!!+4!)/sqr(4) x int(ln(4))
205 = ((4!!)!!+4!)/sqr(4)+int(ln(4))
206 = ((4!!)!!+4!)/sqr(4)+sqr(4)
207 = ((4!!)!!+4!)/sqr(4)+int(ln(4!))
208 = ((4!!)!!+4!)/sqr(4)+4
209 = (((int(ln(4!)))!)!!+4)x4 + int(ln(4))
210 = (((int(ln(4!)))!)!!+4)x4 + sqr(4)
211 = (((int(ln(4!)))!)!!+4)x4 + int(ln(4!))
212 = (((int(ln(4!)))!)!!+4)x4 + 4
213 = (4+int(ln(4!)))!! x sqr(4) + int(ln(4!))
214 = (4+int(ln(4!)))!! x sqr(4) + 4
215 = ((4!!)!!+(int(ln(4!)))!)x4 - int(ln(4))
216 = ((4!!)!!+(int(ln(4!)))!)x4 x int(ln(4))
217 = ((4!!)!!+(int(ln(4!)))!)x4 + int(ln(4))
218 = ((4!!)!!+(int(ln(4!)))!)x4 + sqr(4)
219 = ((4!!)!!+(int(ln(4!)))!)x4 + int(ln(4!))
220 = (((int(ln(4!)))!)!!+4!!)x4 - 4
221 = (((int(ln(4!)))!)!!+4!!)x4 - int(ln(4!))
222 = (((int(ln(4!)))!)!!+4!!)x4 - sqr(4)
223 = (((int(ln(4!)))!)!!+4!!)x4 - int(ln(4))
224 = (((int(ln(4!)))!)!!+4!!)x4 x int(ln(4))
225 = (((int(ln(4!)))!)!!+4!!)x4 + int(ln(4))
226 = (((int(ln(4!)))!)!!+4!!)x4 + sqr(4)
227 = (((int(ln(4!)))!)!!+4!!)x4 + int(ln(4!))
228 = (((int(ln(4!)))!)!!+4!!)x4 + 4
229 = ((4+int(ln(4)))!!)^sqr(4)+4
230 = (4!+4)x4!! + (int(ln(4!)))!
231 = 4^4-4! - int(ln(4))
232 = 4^4-4! x int(ln(4))
233 = 4^4-4! + int(ln(4))
234 = 4^4-4! + sqr(4)
235 = 4^4-4! + int(ln(4!))
236 = 4^4-4! + 4
237 = int(((int(ln(4!)))^sqr(4))!!/4) + int(ln(4))
238 = int(((int(ln(4!)))^sqr(4))!!/4) + sqr(4)
239 = int(((int(ln(4!)))^sqr(4))!!/4) + int(ln(4!))
240 = 4^4-4x4
241 = (int(ln(4!))!)! / int(ln(4!)) +4-int(ln(4!))
242 = (int(ln(4!))!)! / int(ln(4!)) +4-sqr(4)
243 = (int(ln(4!))!)! / int(ln(4!)) +4-int(ln(4))
244 = (int(ln(4!))!)! / int(ln(4!)) +4 x int(ln(4))
245 = (int(ln(4!))!)! / int(ln(4!)) +4+int(ln(4))
246 = (int(ln(4!))!)! / int(ln(4!)) +4+sqr(4)
247 = (int(ln(4!))!)! / int(ln(4!)) +4+int(ln(4!))
248 = (int(ln(4!))!)! / int(ln(4!)) +4+4
249 = (int(ln(4!))!)! / int(ln(4!)) +(int(ln(4!)))^sqr(4)
250 = 4^4-(int(ln(4!)))! x int(ln(4))
251 = 4^4-(int(ln(4!)))! + int(ln(4))
252 = 4^4-(int(ln(4!)))! + sqr(4)
253 = 4^4-(int(ln(4!)))! + int(ln(4!))
254 = 4^4-(int(ln(4!)))! + 4
255 = 4^4-int(ln(4)) x int(ln(4))
256 = 4^4+4-4
257 = 4^4+4-int(ln(4!))
258 = 4^4+4-sqr(4)
259 = 4^4+4-int(ln(4))
260 = 4^4+sqr(4x4)
261 = 4^4+4+int(ln(4))
262 = 4^4+4+sqr(4)
263 = 4^4+4+int(ln(4!))
264 = 4^4+4+4
265 = 4^4+(int(ln(4!)))^sqr(4) 
266 = 4^4+4+(int(ln(4!)))!
267 = 4^4+4!!+int(ln(4!))
268 = 4^4+4!!+4
269 = int((int(ln(4!)))^(4!!)/4!) - 4
270 = int((int(ln(4!)))^(4!!)/4!) - int(ln(4!))
271 = int((int(ln(4!)))^(4!!)/4!) - sqr(4)
272 = int((int(ln(4!)))^(4!!)/4!) - int(ln(4))
273 = int((int(ln(4!)))^(4!!)/4!) x int(ln(4))
274 = int((int(ln(4!)))^(4!!)/4!) + int(ln(4))
275 = int((int(ln(4!)))^(4!!)/4!) + sqr(4)
276 = int((int(ln(4!)))^(4!!)/4!) + int(ln(4!))
277 = int((int(ln(4!)))^(4!!)/4!) + 4
278 = ((int(ln(4!)))!)!! x (int(ln(4!)))! -(int(ln(4!)))! -4
279 = ((int(ln(4!)))!)!! x (int(ln(4!)))! -(int(ln(4!)))^sqr(4)
280 = ((int(ln(4!)))!)!! x (int(ln(4!)))! -4-4
281 = ((int(ln(4!)))!)!! x (int(ln(4!)))! -int(ln(4!)) -4
282 = ((int(ln(4!)))!)!! x (int(ln(4!)))! -sqr(4) -4
283 = ((int(ln(4!)))!)!! x (int(ln(4!)))! -int(ln(4)) -4
284 = ((int(ln(4!)))!)!! x (int(ln(4!)))! -sqr(4x4)
285 = ((int(ln(4!)))!)!! x (int(ln(4!)))! -int(ln(4)) -sqr(4)
286 = ((int(ln(4!)))!)!! x (int(ln(4!)))! -sqr(sqr(4x4))
287 = ((int(ln(4!)))!)!! x (int(ln(4!)))! -4/4
288 = ((int(ln(4!)))!)!! x (int(ln(4!)))! -4+4
289 = ((int(ln(4!)))!)!! x (int(ln(4!)))! +4/4
290 = ((int(ln(4!)))!)!! x (int(ln(4!)))! +sqr(sqr(4x4))
291 = ((int(ln(4!)))!)!! x (int(ln(4!)))! +int(ln(4)) +sqr(4)
292 = ((int(ln(4!)))!)!! x (int(ln(4!)))! +sqr(4x4)
293 = ((int(ln(4!)))!)!! x (int(ln(4!)))! +int(ln(4)) +4
294 = ((int(ln(4!)))!)!! x (int(ln(4!)))! +sqr(4) +4
295 = ((int(ln(4!)))!)!! x (int(ln(4!)))! +int(ln(4!)) +4
296 = ((int(ln(4!)))!)!! x (int(ln(4!)))! +4+4
297 = ((int(ln(4!)))!)!! x (int(ln(4!)))! +(int(ln(4!)))^sqr(4)
298 = ((int(ln(4!)))!)!! x (int(ln(4!)))! +(int(ln(4!)))! +4
299 = (((int(ln(4!)))^sqr(4))!! - ((int(ln(4!)))!)!!)/int(ln(4!))
300 = ((int(ln(4!)))!)!! x (int(ln(4!)))! + (int(ln(4!)))! x sqr(4)
 
Ci saranno sicuramente degli errori.
 
"Sono un po' stanchino"
 
LZ
Ciao

 300 = ((int(ln(4!)))!)!! x (int(ln(4!)))! + (int(ln(4!)))! x sqr(4)
 
300 = 4^4 + 44
 
... ma solo perché mi ci è cascato l'occhio. Degli altri mi fido ciecamente! 
;)
 
Ciao Livio! :)
 
Giorgio 
 


Bello!
Ciao Giorgio!
Livio


Intanto sarei arrivato qua, poi mi fermo per qualche giorno.
 
301 = ((int(ln(4!)))!)!! + sqr(4)) x (int(ln(4!)))! + int(ln(4))
302 = ((int(ln(4!)))!)!! + sqr(4)) x (int(ln(4!)))! + sqr(4)
303 = ((int(ln(4!)))!)!! + sqr(4)) x (int(ln(4!)))! + int(ln(4!))
304 = ((int(ln(4!)))!)!! + sqr(4)) x (int(ln(4!)))! + 4
305 = ((int(ln(4!)))!)!! + int(ln(4!))) x (int(ln(4!)))! - int(ln(4))
306 = ((int(ln(4!)))!)!! + int(ln(4!))) x (int(ln(4!)))! x int(ln(4))
307 = ((int(ln(4!)))!)!! + int(ln(4!))) x (int(ln(4!)))! + int(ln(4))
308 = ((int(ln(4!)))!)!! + int(ln(4!))) x (int(ln(4!)))! + sqr(4)
309 = ((int(ln(4!)))!)!! + int(ln(4!))) x (int(ln(4!)))! + int(ln(4!))
310 = ((int(ln(4!)))!)!! + int(ln(4!))) x (int(ln(4!)))! + 4
311 = ((int(ln(4!)))!)!! + 4) x (int(ln(4!)))! - int(ln(4))
312 = ((int(ln(4!)))!)!! + 4) x (int(ln(4!)))! x int(ln(4))
312 = ((int(ln(4!)))!)!! + 4) x (int(ln(4!)))! + int(ln(4))
312 = ((int(ln(4!)))!)!! + 4) x (int(ln(4!)))! + sqr(4)
312 = ((int(ln(4!)))!)!! + 4) x (int(ln(4!)))! + int(ln(4!))
312 = ((int(ln(4!)))!)!! + 4) x (int(ln(4!)))! + 4
313 = ((int(ln(4!)))^sqr(4))!! / int(ln(4!)) - sqr(4)
314 = ((int(ln(4!)))^sqr(4))!! / int(ln(4!)) - int(ln(4))
315 = ((int(ln(4!)))^sqr(4))!! / int(ln(4!)) x int(ln(4))
316 = ((int(ln(4!)))^sqr(4))!! / int(ln(4!)) + int(ln(4))
317 = ((int(ln(4!)))^sqr(4))!! / int(ln(4!)) + sqr(4)
318 = ((int(ln(4!)))^sqr(4))!! / int(ln(4!)) + int(ln(4!))
319 = ((int(ln(4!)))^sqr(4))!! / int(ln(4!)) + 4
320 = (4!!)!!-4^(int(ln(4!))) x int(ln(4))
321 = (4!!)!!-4^(int(ln(4!))) + int(ln(4))
322 = (4!!)!!-4^(int(ln(4!))) + sqr(4)
323 = (4!!)!!-4^(int(ln(4!))) + int(ln(4!))
324 = (4!!)!!-4^(int(ln(4!))) + 4
325 = (4!!)!! -((int(ln(4!)))!)!! - 4!! - int(ln(4!))
326 = (4!!)!! -((int(ln(4!)))!)!! - 4!! - sqr(4)
327 = (4!!)!! -((int(ln(4!)))!)!! - 4!! - int(ln(4))
328 = (4!!)!! -((int(ln(4!)))!)!! - 4!! x int(ln(4))
329 = (4!!)!! -((int(ln(4!)))!)!! - 4!! + int(ln(4))
330 = (4!!)!! -((int(ln(4!)))!)!! - 4!! + sqr(4)
331 = (4!!)!! -((int(ln(4!)))!)!! - 4!! + int(ln(4!))
332 = (4!!)!! -((int(ln(4!)))!)!! - sqr(4x4)
333 = (4!!)!! -((int(ln(4!)))!)!! - int(ln(4!)) x int(ln(4))
334 = (4!!)!! -((int(ln(4!)))!)!! - sqr(sqr(4x4))
335 = (4!!)!! -((int(ln(4!)))!)!! - int(ln(4)) x int(ln(4))
336 = (4!!)!! -((int(ln(4!)))!)!! - 4+4 
337 = (4!!)!! -((int(ln(4!)))!)!! + 4 -int(ln(4!))
338 = (4!!)!! -((int(ln(4!)))!)!! + 4 -sqr(4) 
339 = (4!!)!! -((int(ln(4!)))!)!! + 4 -int(ln(4)) 
340 = (4!!)!! -((int(ln(4!)))!)!! + 4 x int(ln(4)) 
341 = (4!!)!! -((int(ln(4!)))!)!! + 4 + int(ln(4)) 
342 = (4!!)!! -((int(ln(4!)))!)!! + 4 + sqr(4) 
343 = (4!!)!! -((int(ln(4!)))!)!! + 4 + int(ln(4!)) 
344 = (4!!)!! -((int(ln(4!)))!)!! + 4 + 4 
345 = (4!!)!! -((int(ln(4!)))!)!! + 4!! + int(ln(4))
346 = (4!!)!! -((int(ln(4!)))!)!! + 4!! + sqr(4)
347 = (4!!)!! -((int(ln(4!)))!)!! + 4!! + int(ln(4!))
348 = (4!!)!! -((int(ln(4!)))!)!! + 4!! + 4
349 = (4!!)!! - 4! - 4!! - int(ln(4!))
350 = (4!!)!! - 4! - 4!! - sqr(4)
351 = (4!!)!! - 4! - 4!! - int(ln(4))
352 = (4!!)!! - 4! - 4 - 4
353 = (4!!)!! - 4! - 4 - int(ln(4!))
354 = (4!!)!! - 4! - 4 - sqr(4)
355 = (4!!)!! - 4! - 4 - int(ln(4))
356 = (4!!)!! - 4! - 4 x int(ln(4))
357 = (4!!)!! - 4! - 4 + int(ln(4))
358 = (4!!)!! - 4! - 4 + sqr(4)
359 = (4!!)!! - 4! - 4 + int(ln(4!))
360 = (4!!)!! - 4! + 4 - 4
361 = (4!!)!! - 4! + 4 - int(ln(4!))
362 = (4!!)!! - 4! + 4 - sqr(4)
363 = (4!!)!! - 4! + 4 - int(ln(4))
364 = (4!!)!! - 4! + 4 x int(ln(4))
365 = (4!!)!! - 4! + 4 + int(ln(4))
366 = (4!!)!! - 4! + 4 + sqr(4)
367 = (4!!)!! - 4! + 4 + int(ln(4!))
368 = (4!!)!! - 4! + 4 + 4
369 = (4!!)!! - 4x4 + int(ln(4))
370 = (4!!)!! - 4x4 + sqr(4)
371 = (4!!)!! - 4x4 + int(ln(4!))
372 = (4!!)!! - 4x4 + 4
373 = (4!!)!! - 4!! - 4 + int(ln(4))
374 = (4!!)!! - 4!! - 4 + sqr(4)
375 = (4!!)!! - 4!! - 4 + int(ln(4!))
376 = (4!!)!! - 4!! + 4 - 4
377 = (4!!)!! - 4!! + 4 - int(ln(4!))
378 = (4!!)!! - 4!! + 4 - sqr(4)
379 = (4!!)!! - 4!! + 4 - int(ln(4))
380 = (4!!)!! - 4!! + 4 x int(ln(4))
381 = (4!!)!! - 4!! + 4 + int(ln(4))
382 = (4!!)!! - 4!! + 4 + sqr(4)
383 = (4!!)!! - 4!! + 4 + int(ln(4!))
384 = (4!!)!! - 4!! + 4 + 4
385 = (4!!)!! + 4 - 4 + int(ln(4))
386 = (4!!)!! + 4 - 4 + sqr(4)
387 = (4!!)!! + 4 - 4 + int(ln(4!))
388 = (4!!)!! + 4 + 4 - 4
389 = (4!!)!! + 4 + 4 - int(ln(4!))
390 = (4!!)!! + 4 + 4 - sqr(4)
391 = (4!!)!! + 4 + 4 - int(ln(4))
392 = (4!!)!! + 4 + 4 x int(ln(4!))
393 = (4!!)!! + 4 + 4 + int(ln(4))
394 = (4!!)!! + 4 + 4 + sqr(4)
395 = (4!!)!! + 4 + 4 + int(ln(4!))
396 = (4!!)!! + 4 + 4 + 4
397 = (4!!)!! + 4!! + 4 + int(ln(4))
398 = (4!!)!! + 4!! + 4 + sqr(4)
399 = (4!!)!! + 4!! + 4 + int(ln(4!))
400 = (4!!)!! + 4!! + 4!! x int(ln(4))
401 = (4!!)!! + 4!! + 4!! + int(ln(4))
402 = (4!!)!! + 4!! + 4!! + sqr(4)
403 = (4!!)!! + 4!! + 4!! + int(ln(4!))
404 = (4!!)!! + 4!! + 4!! + 4
405 = (4!!)!! + 4! - 4 + int(ln(4))
406 = (4!!)!! + 4! - 4 + sqr(4)
407 = (4!!)!! + 4! - 4 + int(ln(4!))
408 = (4!!)!! + 4! - 4 + 4
409 = (4!!)!! + 4! + 4 - int(ln(4!))
410 = (4!!)!! + 4! + 4 - sqr(4)
411 = (4!!)!! + 4! + 4 - int(ln(4))
412 = (4!!)!! + 4! + 4 x int(ln(4))
413 = (4!!)!! + 4! + 4 + int(ln(4))
414 = (4!!)!! + 4! + 4 + sqr(4)
415 = (4!!)!! + 4! + 4 + int(ln(4!))
416 = (4!!)!! + 4! + 4 + 4
417 = (4!!)!! + 4! + 4!! + int(ln(4))
418 = (4!!)!! + 4! + 4!! + sqr(4)
419 = (4!!)!! + 4! + 4!! + int(ln(4!))
420 = (4!!)!! + 4! + 4!! + 4
421 = (4!!)!! + ((int(ln(4!)))!)!! - 4!! - int(ln(4!))
422 = (4!!)!! + ((int(ln(4!)))!)!! - 4!! - sqr(4)
423 = (4!!)!! + ((int(ln(4!)))!)!! - 4!! - int(ln(4))
424 = (4!!)!! + ((int(ln(4!)))!)!! - 4!! x int(ln(4))
425 = (4!!)!! + ((int(ln(4!)))!)!! - 4!! + int(ln(4))
426 = (4!!)!! + ((int(ln(4!)))!)!! - 4!! + sqr(4)
427 = (4!!)!! + ((int(ln(4!)))!)!! - 4!! + int(ln(4!))
428 = (4!!)!! + ((int(ln(4!)))!)!! - 4!! + 4
429 = (4!!)!! + ((int(ln(4!)))!)!! - 4 + int(ln(4))
430 = (4!!)!! + ((int(ln(4!)))!)!! - 4 + sqr(4)
431 = (4!!)!! + ((int(ln(4!)))!)!! - 4 + int(ln(4!))
432 = (4!!)!! + ((int(ln(4!)))!)!! - 4 + 4
433 = (4!!)!! + ((int(ln(4!)))!)!! + 4 - int(ln(4!))
434 = (4!!)!! + ((int(ln(4!)))!)!! + 4 - sqr(4)
435 = (4!!)!! + ((int(ln(4!)))!)!! + 4 - int(ln(4))
436 = (4!!)!! + ((int(ln(4!)))!)!! + 4 x int(ln(4))
437 = (4!!)!! + ((int(ln(4!)))!)!! + 4 + int(ln(4))
438 = (4!!)!! + ((int(ln(4!)))!)!! + 4 + sqr(4)
439 = (4!!)!! + ((int(ln(4!)))!)!! + 4 + int(ln(4!))
440 = 444 - 4
441 = 444 - int(ln(4!))
442 = 444 - sqr(4)
443 = 444 - int(ln(4))
444 = 444 x int(ln(4))
445 = 444 + int(ln(4))
446 = 444 + sqr(4)
447 = 444 + int(ln(4!))
448 = 444 + 4
449 = (4!!)!! + 4^int(ln(4!)) + int(ln(4))
450 = (4!!)!! + 4^int(ln(4!)) + sqr(4)
451 = (4!!)!! + 4^int(ln(4!)) + int(ln(4!))
452 = (4!!)!! + 4^int(ln(4!)) + 4
453 = (4!!)!! + S((int(ln(4!)))!,int(ln(4))) + 4
454 = (4!!)!! + 4^int(ln(4!)) + (int(ln(4!)))!
455 = (4!!)!! + S((int(ln(4!)))!,int(ln(4))) + (int(ln(4!)))!
456 = (4!!)!! + 4^int(ln(4!)) + 4!!
457 = (4!!)!! + S((int(ln(4!)))!,int(ln(4))) + 4!!
458 = S((int(ln(4!)))^sqr(4),int(ln(4))) - 4
459 = S((int(ln(4!)))^sqr(4),int(ln(4))) - int(ln(4!))
460 = S((int(ln(4!)))^sqr(4),int(ln(4))) - sqr(4)
461 = S((int(ln(4!)))^sqr(4),int(ln(4))) - int(ln(4))
462 = S((int(ln(4!)))^sqr(4),int(ln(4))) x int(ln(4))
463 = S((int(ln(4!)))^sqr(4),int(ln(4))) + int(ln(4))
464 = (((int(ln(4!)))!)!! + int(ln((4!!)!))) x 4!! x int(ln(4))
465 = (((int(ln(4!)))!)!! + int(ln((4!!)!))) x 4!! + int(ln(4))
466 = (((int(ln(4!)))!)!! + int(ln((4!!)!))) x 4!! + sqr(4)
467 = (((int(ln(4!)))!)!! + int(ln((4!!)!))) x 4!! + int(ln(4!))
468 = (((int(ln(4!)))!)!! + int(ln((4!!)!))) x 4!! + 4
469 = (((int(ln(4!)))!)!! + int(ln((4!!)!))) x 4!! + int(ln((4!!)!!))
470 = (((int(ln(4!)))!)!! + int(ln((4!!)!))) x 4!! + (int(ln(4!)))!
471 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) - (int(ln(4!)))^sqr(4)
472 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) - 4-4
473 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) - 4-int(ln(4!))
474 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) - (int(ln(4!)))! x int(ln(4))
475 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) - 4-int(ln(4))
476 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) - sqr(4x4)
477 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) - int(ln(4!)) x int(ln(4))
478 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) - sqr(sqr(4x4))
479 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) - 4/4
480 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + 4-4
481 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + 4/4
482 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + sqr(sqr(4x4))
483 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + int(ln(4!)) x int(ln(4))
484 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + sqr(4x4)
485 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + 4+int(ln(4))
486 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + (int(ln(4!)))! x int(ln(4))
487 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + 4+int(ln(4!))
488 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + 4+4
489 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + (int(ln(4!)))^sqr(4)
490 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + 4+(int(ln(4!)))!
491 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + int(ln((4!!)!!))+(int(ln(4!)))!
492 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + 4 x int(ln(4!))
493 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + 4!! x int(ln((4!!)!!))
494 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + 4!! x (int(ln(4!)))!
495 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + int(ln((4!!)!)) + int(ln((4!!)!!))
496 = ((int(ln(4!)))!)!! x int(ln((4!!)!)) + 4x4
497 = int(ln(sqr(sqr(sqr((((int(ln(4!)))!)!!)!))))) x (4!+(int(ln(4!)))!) - int(ln(sqr(sqr((4!)!))))
 
 
NOTE:
x!!     semifattoriale
S(n,k)  numeri di Stirling di seconda specie
int     funzione parte intera
abs     valore assoluto
ln      logaritmo naturale
 
65 = S(6,1) numero di Stirling
462 = S(9,1) numero di Stirling
 
1 = int(ln(4))
2 = sqr(4)
3 = int(ln(4!))
4 = 4
5 = int(ln((4!!)!!))
6 = (int(ln(4!)))!
7 = 4+int(ln(4!))
8 = 4!!
9 = (int(ln(4!)))^sqr(4)
10 = int(ln((4!!)!))
 
13 = int(ln(sqr(sqr((4!)!))))
 
17 = int(ln(sqr(sqr(sqr((((int(ln(4!)))!)!!)!)))))
 
 
24 = 4!
48 = (int(ln(4!)))!)!!
384 = (4!!)!!
40320 = (4!!)!

 414 = (4!!)!! + 4! + 4 + sqr(4)
 
Mi sa che non sara' per niente facile ne' piccolo, perche' vedi: il 414
lo becchi in un sol colpo cosi':
 
int(Log(sf(sf(sf(4)))))
 
essendo Log() il log in base 10, che come funzione non puo' ch'essere
valida al pari del log naturale ln() (che in mat e fis s'usa scrivere
ancora log() invece di ln()) e che in ogni caso figura col simbolo log()
(minuscolo) in tutte le calcolatrici ciao
 
-- 
Yoda
 

 int(Log(sf(sf(sf(4)))))
 
Scusa sf() e' !!, ho incollato dalla riga di comando di bc, per la quale
ho fatto la funzione chiamandola sf(), eccola ciao
 
/* semifattoriale recorsivo */
define sf(x) {
if (x<=1) return (1);
return (x * sf(x-2));
}
 
-- 
Yoda
 

 Quale sara' il primo intero non rappresentabile da quattro 4?
 
La pagina di Wikipedia dedicata al puzzle
(
https://en.wikipedia.org/wiki/Four_fours
) 
fa notare che, se si ammette il logaritmo fra i simboli da usare, allora
si può trovare qualunque numero così:
 
n = -√(4) ln[ln(√√√√...√4)/ln(4)]/ln(4)
 
dove √ è il simbolo di radice quadrata (spero che compaia giusto) e ad
argomento del ln ci sono n radici quadrate annidate.
 
M.
 
-- 
Per rispondermi via e-mail togli l'ovvio.


Grazie!
Almeno sospendo la ricerca! :-)
Forse potrebbe restare in piedi la domanda di qual e' il primo intero non rappresentabile da quattro 4 senza usare i logaritmi, oppure usando ben determinate operazioni matematiche.

 ben determinate operazioni matematiche.
 
con le 4 aritmetiche forse diventa banale, magari invece aggiungendo
le potenze ad esponente intero? ciao
 
-- 
Yoda
 


33 = 4!+!4x4/4
34 = 4!+4+4+sqr(4) 
35 = 4!+4/sqr(4)+!4 
36 = 44-4-4 
37 = 44-!4+sqr(4) 
38 = 44-4-sqr(4) 
39 = gamma(4) x gamma(4) + gamma(4) / sqr(4)
40 = 44-sqr(4)-sqr(4) 
41 = gamma(4) x gamma(4) + sqr(4) + sqr(!4)
42 = 44-4/sqr(4) 
43 = 44-4/4 
44 = 44+4-4 
45 = 44+4/4 
46 = 44+4/sqr(4)
47 = gamma(4) x gamma(4) + sqr(4) + !4
48 = gamma(4) x 4!! x 4/4
49 = gamma(4) x 4!! + 4/4
...
 
La tentazione è di nuovo forte. :-)