Primes: n*2^m+1

From Rosetta Code
Task
Primes: n*2^m+1
You are encouraged to solve this task according to the task description, using any language you may know.
Task
  • Find and display the first 45 (n) primes of the form n × 2m + 1 where m is the smallest valid non-negative integer.


Stretch
  • Find and display the first 50 (n) primes of the form n × 2m + 1 where m is the smallest valid non-negative integer.


Stretch harder
  • Find and display the first 400 (n) primes of the form n × 2m + 1 where m is the smallest valid non-negative integer. Specifically term 383.


See also

A050921 - Smallest prime of form n*2^m+1


ALGOL 68

Works with: ALGOL 68G version Any - Tested with release 3.0.3 under Windows

Handles the stretchier stretch goal, though you will have to wait a while...
Most of the primes will fit in 64 bits (those up to 45 will fit in 16 bits) but there are a small number that have hundreds or thousands of digits.
NB the primes.incl.a68 source is available on a page in Rosetta Code - see the library above.

BEGIN # find primes of the form 1+n*2^m where m is the lowest integer >= 0   #
      # such that 1+n*2^m is prime                                           #
    PR read "primes.incl.a68" PR # include peime utilities                   #
    PR precision 8000 PR # set the precision of LONG LONG INT                #
    INT max m := 8000;   # maximum m we will consider                        #
    FOR n TO 400 DO
        INT           m         := 0;
        LONG LONG INT nx2 to m  := n;
        LONG LONG INT p         := 0;
        BOOL          found     := FALSE;
        WHILE NOT found AND m <= max m DO
            IF NOT ( found := is probably prime( p := nx2 to m + 1 ) ) THEN
                nx2 to m *:= 2;
                m        +:= 1
            FI
        OD;
        IF NOT found THEN
            print( ( whole( n, -3 ), " not found", newline ) )
        ELSE
            print( ( whole( n, -3 ), " ", whole( m, -8 ), ": ", whole( p, 0 ), newline ) )
        FI
    OD
END
Output:
  1        0: 2
  2        0: 3
  3        1: 7
  4        0: 5
  5        1: 11
  6        0: 7
  7        2: 29
  8        1: 17
  9        1: 19
 10        0: 11
 11        1: 23
 12        0: 13
 13        2: 53
 14        1: 29
 15        1: 31
 16        0: 17
 17        3: 137
 18        0: 19
 19        6: 1217
 20        1: 41
 21        1: 43
 22        0: 23
 23        1: 47
 24        2: 97
 25        2: 101
 26        1: 53
 27        2: 109
 28        0: 29
 29        1: 59
 30        0: 31
 31        8: 7937
 32        3: 257
 33        1: 67
 34        2: 137
 35        1: 71
 36        0: 37
 37        2: 149
 38        5: 1217
 39        1: 79
 40        0: 41
 41        1: 83
 42        0: 43
 43        2: 173
 44        1: 89
 45        2: 181
 46        0: 47
 47      583: 1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
 48        1: 97
 49        2: 197
 50        1: 101
 51        1: 103
 52        0: 53
 53        1: 107
 54        1: 109
 55        4: 881
 56        1: 113
 57        2: 229
 58        0: 59
 59        5: 1889
 60        0: 61
 61        4: 977
 62        7: 7937
 63        1: 127
 64        2: 257
 65        1: 131
 66        0: 67
 67        2: 269
 68        1: 137
 69        1: 139
 70        0: 71
 71        3: 569
 72        0: 73
 73        2: 293
 74        1: 149
 75        1: 151
 76        4: 1217
 77        3: 617
 78        0: 79
 79        2: 317
 80        3: 641
 81        1: 163
 82        0: 83
 83        1: 167
 84        2: 337
 85        4: 1361
 86        1: 173
 87        2: 349
 88        0: 89
 89        1: 179
 90        1: 181
 91        8: 23297
 92        7: 11777
 93        2: 373
 94      582: 1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
 95        1: 191
 96        0: 97
 97        2: 389
 98        1: 197
 99        1: 199
100        0: 101
101        3: 809
102        0: 103
103       16: 6750209
104        5: 3329
105        1: 211
106        0: 107
107        3: 857
108        0: 109
109        6: 6977
110        3: 881
111        1: 223
112        0: 113
113        1: 227
114        1: 229
115        2: 461
116        1: 233
117        3: 937
118        4: 1889
119        1: 239
120        1: 241
121        8: 30977
122        3: 977
123        6: 7873
124        6: 7937
125        1: 251
126        0: 127
127        2: 509
128        1: 257
129        3: 1033
130        0: 131
131        1: 263
132        4: 2113
133        4: 2129
134        1: 269
135        1: 271
136        0: 137
137        3: 1097
138        0: 139
139        2: 557
140        1: 281
141        1: 283
142        2: 569
143       53: 1288029493427961857
144        2: 577
145        6: 9281
146        1: 293
147        8: 37633
148        0: 149
149        3: 1193
150        0: 151
151        4: 2417
152        3: 1217
153        1: 307
154        2: 617
155        1: 311
156        0: 157
157        8: 40193
158        1: 317
159        6: 10177
160        2: 641
161        3: 1289
162        0: 163
163        2: 653
164        9: 83969
165        1: 331
166        0: 167
167        7: 21377
168        1: 337
169        2: 677
170        3: 1361
171        8: 43777
172        0: 173
173        1: 347
174        1: 349
175        2: 701
176        1: 353
177        2: 709
178        0: 179
179        1: 359
180        0: 181
181        4: 2897
182        7: 23297
183        1: 367
184        6: 11777
185        3: 1481
186        1: 373
187        6: 11969
188      581: 1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
189        1: 379
190        0: 191
191        1: 383
192        0: 193
193        2: 773
194        1: 389
195        4: 3121
196        0: 197
197       15: 6455297
198        0: 199
199        2: 797
200        1: 401
201        3: 1609
202        2: 809
203       13: 1662977
204        1: 409
205        2: 821
206       15: 6750209
207        2: 829
208        4: 3329
209        1: 419
210        0: 211
211       20: 221249537
212        3: 1697
213        2: 853
214        2: 857
215        1: 431
216        1: 433
217       66: 16011773855979890802689
218        5: 6977
219        1: 439
220        2: 881
221        1: 443
222        0: 223
223        8: 57089
224        1: 449
225        3: 1801
226        0: 227
227       11: 464897
228        0: 229
229        6: 14657
230        1: 461
231        1: 463
232        0: 233
233        1: 467
234        2: 937
235        2: 941
236        3: 1889
237        4: 3793
238        0: 239
239        1: 479
240        0: 241
241       36: 16561393893377
242        7: 30977
243        1: 487
244        2: 977
245        1: 491
246        5: 7873
247        6: 15809
248        5: 7937
249        1: 499
250        0: 251
251        1: 503
252        2: 1009
253        2: 1013
254        1: 509
255        2: 1021
256        0: 257
257      279: 249632952651006185613150855026822179503549278818199928480857894651449200648869292015617
258        2: 1033
259       38: 71193377898497
260        1: 521
261        1: 523
262        0: 263
263       29: 141197049857
264        3: 2113
265        2: 1061
266        3: 2129
267        2: 1069
268        0: 269
269        3: 2153
270        0: 271
271        4: 4337
272       11: 557057
273        1: 547
274        2: 1097
275        7: 35201
276        0: 277
277        2: 1109
278        1: 557
279        2: 1117
280        0: 281
281        1: 563
282        0: 283
283       30: 303868936193
284        1: 569
285        1: 571
286       52: 1288029493427961857
287        3: 2297
288        1: 577
289       10: 295937
290        5: 9281
291        4: 4657
292        0: 293
293        1: 587
294        7: 37633
295        2: 1181
296        1: 593
297        3: 2377
298        2: 1193
299        1: 599
300        1: 601
301        4: 4817
302        3: 2417
303        1: 607
304        2: 1217
305        3: 2441
306        0: 307
307        2: 1229
308        1: 617
309        1: 619
310        0: 311
311        9: 159233
312        0: 313
313        4: 5009
314        7: 40193
315        1: 631
316        0: 317
317        7: 40577
318        5: 10177
319        2: 1277
320        1: 641
321        1: 643
322        2: 1289
323        1: 647
324        2: 1297
325        2: 1301
326        1: 653
327        3: 2617
328        8: 83969
329        1: 659
330        0: 331
331        4: 5297
332        3: 2657
333        5: 10657
334        6: 21377
335       19: 175636481
336        0: 337
337        4: 5393
338        1: 677
339        3: 2713
340        2: 1361
341        1: 683
342        7: 43777
343        2: 1373
344        3: 2753
345        1: 691
346        0: 347
347        3: 2777
348        0: 349
349       10: 357377
350        1: 701
351       12: 1437697
352        0: 353
353       21: 740294657
354        1: 709
355        6: 22721
356        5: 11393
357        2: 1429
358        0: 359
359        1: 719
360        6: 23041
361       28: 96905199617
362        3: 2897
363        1: 727
364        6: 23297
365        5: 11681
366        0: 367
367       12: 1503233
368        5: 11777
369        1: 739
370        2: 1481
371        1: 743
372        0: 373
373        2: 1493
374        5: 11969
375        1: 751
376      580: 1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
377       11: 772097
378        0: 379
379       14: 6209537
380        1: 761
381        3: 3049
382        0: 383
383     6393: 11693945185971565896920916176753769281418376445302724140914106576604960252116205468905429628661873192664799900323401294531072465400997845029722990758855393414014415817179228695517839305455702961095094596926622802342799137107509767542153683280899327558274011281588755909890607960835140712630830933978801393590855371457894042968287926562847826310125559303901351824980311279986492793008248059208985097459095049075732193161126922389950080848742183055141518931962329796357335158955758486061360294773463111842316561192036096585088267052290025273980611139612478214293303564141730470933187279751846912161098280963960686648202780382930927114525552446602357404550468641236474238897222372272898562140228039886991631673186995098587756569010989657598363351856992206826342175536967926902668804937341514786382018872919876784539436965319822540039220122728568129762675989071883516915894567537630751801497223803135172643203770169327233350522822938630733126833423559124391441973547309619943019237705312515304113424366223388373606440335025932390399945086075175009569272136997988977568262327875607690344516747889133920438003737328060362069562108376086129279385800262195985974144460914705464874882401864174074796383557151951711000378565395148939760434428093058777242253682181813425273399277638142811972296863003382484684788329148214958434057306251885787781329925372401240556666727438408378656900945061970219566055969587385482421092779185798692904507774583223151161566406541599486350580593707153172641891804260963429951215526999443852964537303345106153870841180251403751871193132336680841124129779119999935597712685839886558769823834654994044516702436738265181698869580022472787153167463772595005393815295009535991557511340157179280662197799109181549751673455040271529561595718940092424231253150263268513067972937042222806102175350331146290864120703025608712817763221723427454002746818270565050919821097445991953785331131470462682015972815241620750337
384        1: 769
385        8: 98561
386        1: 773
387        2: 1549
388        0: 389
389       11: 796673
390        3: 3121
391        4: 6257
392        3: 3137
393        1: 787
394       14: 6455297
395        5: 12641
396        0: 397
397        4: 6353
398        1: 797
399        2: 1597
400        0: 401

ALGOL W

Although most of the primes up to 500 will fit in 32 bits, obviously 383 won't, having over 6000 digits, so this doesn;t attempt to show more than the first 45. NB: 47 is the first prime that won't fit in 32 bits.

begin % find primes of the form 1+n*2^m where m is the lowest integer >= 0   %
      % such that 1+n*2^m is prime                                           %
    integer MAX_M, MAX_PRIME;
    MAX_M     := 22;
    MAX_PRIME := 10000;
    begin
        logical array prime ( 1 :: MAX_PRIME );
        % sieve the primes to MAX_PRIME                                      %
        prime( 1 ) := false; prime( 2 ) := true;
        for i := 3 step 2 until MAX_PRIME do prime( i ) := true;
        for i := 4 step 2 until MAX_PRIME do prime( i ) := false;
        for i := 3 step 2 until truncate( sqrt( MAX_PRIME ) ) do begin
            integer ii; ii := i + i;
            if prime( i ) then for pr := i * i step ii until MAX_PRIME do prime( pr ) := false
        end for_i ;
        % find the n*2^m + 1 primes                                          %
        for n := 1 until 45 do begin
            integer m, twoToM, p;
            logical notFound;
            m        := 0;
            twoToM   := 1;
            p        := 0;
            notFound := true;
            while notFound and m <= MAX_M do begin
                p := ( n * twoToM ) + 1;
                notFound := not prime( p );
                if notFound then begin
                    twoToM := twoToM + twoToM;
                    m      := m + 1
                end if_notFound
            end while_notFound_and_m_le_MAX_M ;
            if notFound
            then writeon( i_w := 3, s_w := 0, "(", n, " not found)" )
            else writeon( i_w := 3, s_w := 0, "(", n, " ", i_w := 1, m, ": ", i_w := 4, p, "  )" );
            if n rem 5 = 0 then write()
        end for_n
    end
end.
Output:
(  1 0:    2  )(  2 0:    3  )(  3 1:    7  )(  4 0:    5  )(  5 1:   11  )
(  6 0:    7  )(  7 2:   29  )(  8 1:   17  )(  9 1:   19  )( 10 0:   11  )
( 11 1:   23  )( 12 0:   13  )( 13 2:   53  )( 14 1:   29  )( 15 1:   31  )
( 16 0:   17  )( 17 3:  137  )( 18 0:   19  )( 19 6: 1217  )( 20 1:   41  )
( 21 1:   43  )( 22 0:   23  )( 23 1:   47  )( 24 2:   97  )( 25 2:  101  )
( 26 1:   53  )( 27 2:  109  )( 28 0:   29  )( 29 1:   59  )( 30 0:   31  )
( 31 8: 7937  )( 32 3:  257  )( 33 1:   67  )( 34 2:  137  )( 35 1:   71  )
( 36 0:   37  )( 37 2:  149  )( 38 5: 1217  )( 39 1:   79  )( 40 0:   41  )
( 41 1:   83  )( 42 0:   43  )( 43 2:  173  )( 44 1:   89  )( 45 2:  181  )

Arturo

cnt: 0
n: 1
while [cnt < 45][
    m: 0
    while [true][
        p: inc n * 2^m
        if prime? p [
            print ["n:" n "m:" m "p:" p]
            inc 'cnt
            break
        ]
        inc 'm
    ]
    inc 'n
]
Output:
n: 1 m: 0 p: 2 
n: 2 m: 0 p: 3 
n: 3 m: 1 p: 7 
n: 4 m: 0 p: 5 
n: 5 m: 1 p: 11 
n: 6 m: 0 p: 7 
n: 7 m: 2 p: 29 
n: 8 m: 1 p: 17 
n: 9 m: 1 p: 19 
n: 10 m: 0 p: 11 
n: 11 m: 1 p: 23 
n: 12 m: 0 p: 13 
n: 13 m: 2 p: 53 
n: 14 m: 1 p: 29 
n: 15 m: 1 p: 31 
n: 16 m: 0 p: 17 
n: 17 m: 3 p: 137 
n: 18 m: 0 p: 19 
n: 19 m: 6 p: 1217 
n: 20 m: 1 p: 41 
n: 21 m: 1 p: 43 
n: 22 m: 0 p: 23 
n: 23 m: 1 p: 47 
n: 24 m: 2 p: 97 
n: 25 m: 2 p: 101 
n: 26 m: 1 p: 53 
n: 27 m: 2 p: 109 
n: 28 m: 0 p: 29 
n: 29 m: 1 p: 59 
n: 30 m: 0 p: 31 
n: 31 m: 8 p: 7937 
n: 32 m: 3 p: 257 
n: 33 m: 1 p: 67 
n: 34 m: 2 p: 137 
n: 35 m: 1 p: 71 
n: 36 m: 0 p: 37 
n: 37 m: 2 p: 149 
n: 38 m: 5 p: 1217 
n: 39 m: 1 p: 79 
n: 40 m: 0 p: 41 
n: 41 m: 1 p: 83 
n: 42 m: 0 p: 43 
n: 43 m: 2 p: 173 
n: 44 m: 1 p: 89 
n: 45 m: 2 p: 181

EasyLang

Translation of: FreeBASIC
func isprim num .
   i = 2
   while i <= sqrt num
      if num mod i = 0
         return 0
      .
      i += 1
   .
   return 1
.
for n = 1 to 45
   m = 0
   repeat
      p = n * (pow 2 m) + 1
      until isprim p = 1
      m += 1
   .
   print n & " " & m & " " & p
.


FreeBASIC

#include "isprime.bas"

Print !"  N     M    Prime\n------------------"
For n As Integer = 1 To 45
    Dim As Ulongint m = 0
    Do
        Dim As Ulongint p = n * (2 ^ m) + 1
        If isPrime(p) Then
            Print Using "###  ####   ####"; n; m; p
            Exit Do
        End If
        m += 1
    Loop
Next n
Sleep

J

   ' n m prime',":(,.1+(*2^])/@|:)(,.~#\)i.&1"1]1 p:1+(1+i.45) */ 2^i.9
 n m prime
 1 0    2 
 2 0    3 
 3 1    7 
 4 0    5 
 5 1   11 
 6 0    7 
 7 2   29 
 8 1   17 
 9 1   19 
10 0   11 
11 1   23 
12 0   13 
13 2   53 
14 1   29 
15 1   31 
16 0   17 
17 3  137 
18 0   19 
19 6 1217 
20 1   41 
21 1   43 
22 0   23 
23 1   47 
24 2   97 
25 2  101 
26 1   53 
27 2  109 
28 0   29 
29 1   59 
30 0   31 
31 8 7937 
32 3  257 
33 1   67 
34 2  137 
35 1   71 
36 0   37 
37 2  149 
38 5 1217 
39 1   79 
40 0   41 
41 1   83 
42 0   43 
43 2  173 
44 1   89 
45 2  181

(Most of the implementation here is about merging intermediate values and formatting for display. The calculation for m is i.&1"1]1 p:1+(1+i.45) */ 2^i.9 -- for n in the range 1..45, try all m exponents in the range 0..8 and find the first m value for each n which corresponds to a prime.)

Java

Takes about 10 minutes to find the primes for n up to 400 on my Windows 11 laptop. All but a few seconds of that time is spent finding 383.
Tested with OpenJDK version 22.

import java.math.BigInteger;
public class primesNx2ToMPlus1 // find primes of the form 1+n*2^m where m is
{                              // the lowest integer >= 0 such that 1+n*2^m is prime
    static final int maxM = 8000;   // maximum m we will consider
    public static void main( String[] args )
    {
        BigInteger nn = BigInteger.ZERO;
        for( int n = 1; n <= 400; n ++ )
        {
            int        m        = 0;
            boolean    found    = false;
            nn                  = nn.add( BigInteger.ONE );
            BigInteger nx2ToM   = nn;
            BigInteger p        = BigInteger.ZERO;
            while( ! found && m <= maxM )
            {
                p = nx2ToM.add( BigInteger.ONE );
                if( ! ( found = p.isProbablePrime( 10 ) ) )
                {
                    nx2ToM = nx2ToM.add( nx2ToM );
                    m     += 1;
                }
            }
            System.out.print( String.format( "%3d", n ) );
            if( ! found )
            {
                System.out.println( " not found" );
            }
            else
            {
                System.out.println( " " + String.format( "%6d", m ) + ": " + p.toString() );
            }
        }
    } // main
} // primesNx2ToMPlus1
Output:

As with many other samples, the long primes have been manually shortened.

  1      0: 2
  2      0: 3
  3      1: 7
  4      0: 5
  5      1: 11
  6      0: 7
  7      2: 29
  8      1: 17
  9      1: 19
 10      0: 11
 11      1: 23
 12      0: 13
 13      2: 53
 14      1: 29
 15      1: 31
 16      0: 17
 17      3: 137
 18      0: 19
 19      6: 1217
 20      1: 41
 21      1: 43
 22      0: 23
 23      1: 47
 24      2: 97
 25      2: 101
 26      1: 53
 27      2: 109
 28      0: 29
 29      1: 59
 30      0: 31
 31      8: 7937
 32      3: 257
 33      1: 67
 34      2: 137
 35      1: 71
 36      0: 37
 37      2: 149
 38      5: 1217
 39      1: 79
 40      0: 41
 41      1: 83
 42      0: 43
 43      2: 173
 44      1: 89
 45      2: 181
 46      0: 47
 47    583: 148793969526...246574002177
 48      1: 97
 49      2: 197
 50      1: 101
 51      1: 103
 52      0: 53
 53      1: 107
 54      1: 109
 55      4: 881
 56      1: 113
 57      2: 229
 58      0: 59
 59      5: 1889
 60      0: 61
 61      4: 977
 62      7: 7937
 63      1: 127
 64      2: 257
 65      1: 131
 66      0: 67
 67      2: 269
 68      1: 137
 69      1: 139
 70      0: 71
 71      3: 569
 72      0: 73
 73      2: 293
 74      1: 149
 75      1: 151
 76      4: 1217
 77      3: 617
 78      0: 79
 79      2: 317
 80      3: 641
 81      1: 163
 82      0: 83
 83      1: 167
 84      2: 337
 85      4: 1361
 86      1: 173
 87      2: 349
 88      0: 89
 89      1: 179
 90      1: 181
 91      8: 23297
 92      7: 11777
 93      2: 373
 94    582: 148793969526...246574002177
 95      1: 191
 96      0: 97
 97      2: 389
 98      1: 197
 99      1: 199
100      0: 101
101      3: 809
102      0: 103
103     16: 6750209
104      5: 3329
105      1: 211
106      0: 107
107      3: 857
108      0: 109
109      6: 6977
110      3: 881
111      1: 223
112      0: 113
113      1: 227
114      1: 229
115      2: 461
116      1: 233
117      3: 937
118      4: 1889
119      1: 239
120      1: 241
121      8: 30977
122      3: 977
123      6: 7873
124      6: 7937
125      1: 251
126      0: 127
127      2: 509
128      1: 257
129      3: 1033
130      0: 131
131      1: 263
132      4: 2113
133      4: 2129
134      1: 269
135      1: 271
136      0: 137
137      3: 1097
138      0: 139
139      2: 557
140      1: 281
141      1: 283
142      2: 569
143     53: 1288029493427961857
144      2: 577
145      6: 9281
146      1: 293
147      8: 37633
148      0: 149
149      3: 1193
150      0: 151
151      4: 2417
152      3: 1217
153      1: 307
154      2: 617
155      1: 311
156      0: 157
157      8: 40193
158      1: 317
159      6: 10177
160      2: 641
161      3: 1289
162      0: 163
163      2: 653
164      9: 83969
165      1: 331
166      0: 167
167      7: 21377
168      1: 337
169      2: 677
170      3: 1361
171      8: 43777
172      0: 173
173      1: 347
174      1: 349
175      2: 701
176      1: 353
177      2: 709
178      0: 179
179      1: 359
180      0: 181
181      4: 2897
182      7: 23297
183      1: 367
184      6: 11777
185      3: 1481
186      1: 373
187      6: 11969
188    581: 148793969526...246574002177
189      1: 379
190      0: 191
191      1: 383
192      0: 193
193      2: 773
194      1: 389
195      4: 3121
196      0: 197
197     15: 6455297
198      0: 199
199      2: 797
200      1: 401
201      3: 1609
202      2: 809
203     13: 1662977
204      1: 409
205      2: 821
206     15: 6750209
207      2: 829
208      4: 3329
209      1: 419
210      0: 211
211     20: 221249537
212      3: 1697
213      2: 853
214      2: 857
215      1: 431
216      1: 433
217     66: 16011773855979890802689
218      5: 6977
219      1: 439
220      2: 881
221      1: 443
222      0: 223
223      8: 57089
224      1: 449
225      3: 1801
226      0: 227
227     11: 464897
228      0: 229
229      6: 14657
230      1: 461
231      1: 463
232      0: 233
233      1: 467
234      2: 937
235      2: 941
236      3: 1889
237      4: 3793
238      0: 239
239      1: 479
240      0: 241
241     36: 16561393893377
242      7: 30977
243      1: 487
244      2: 977
245      1: 491
246      5: 7873
247      6: 15809
248      5: 7937
249      1: 499
250      0: 251
251      1: 503
252      2: 1009
253      2: 1013
254      1: 509
255      2: 1021
256      0: 257
257    279: 249632952651...869292015617
258      2: 1033
259     38: 71193377898497
260      1: 521
261      1: 523
262      0: 263
263     29: 141197049857
264      3: 2113
265      2: 1061
266      3: 2129
267      2: 1069
268      0: 269
269      3: 2153
270      0: 271
271      4: 4337
272     11: 557057
273      1: 547
274      2: 1097
275      7: 35201
276      0: 277
277      2: 1109
278      1: 557
279      2: 1117
280      0: 281
281      1: 563
282      0: 283
283     30: 303868936193
284      1: 569
285      1: 571
286     52: 1288029493427961857
287      3: 2297
288      1: 577
289     10: 295937
290      5: 9281
291      4: 4657
292      0: 293
293      1: 587
294      7: 37633
295      2: 1181
296      1: 593
297      3: 2377
298      2: 1193
299      1: 599
300      1: 601
301      4: 4817
302      3: 2417
303      1: 607
304      2: 1217
305      3: 2441
306      0: 307
307      2: 1229
308      1: 617
309      1: 619
310      0: 311
311      9: 159233
312      0: 313
313      4: 5009
314      7: 40193
315      1: 631
316      0: 317
317      7: 40577
318      5: 10177
319      2: 1277
320      1: 641
321      1: 643
322      2: 1289
323      1: 647
324      2: 1297
325      2: 1301
326      1: 653
327      3: 2617
328      8: 83969
329      1: 659
330      0: 331
331      4: 5297
332      3: 2657
333      5: 10657
334      6: 21377
335     19: 175636481
336      0: 337
337      4: 5393
338      1: 677
339      3: 2713
340      2: 1361
341      1: 683
342      7: 43777
343      2: 1373
344      3: 2753
345      1: 691
346      0: 347
347      3: 2777
348      0: 349
349     10: 357377
350      1: 701
351     12: 1437697
352      0: 353
353     21: 740294657
354      1: 709
355      6: 22721
356      5: 11393
357      2: 1429
358      0: 359
359      1: 719
360      6: 23041
361     28: 96905199617
362      3: 2897
363      1: 727
364      6: 23297
365      5: 11681
366      0: 367
367     12: 1503233
368      5: 11777
369      1: 739
370      2: 1481
371      1: 743
372      0: 373
373      2: 1493
374      5: 11969
375      1: 751
376    580: 148793969526...246574002177
377     11: 772097
378      0: 379
379     14: 6209537
380      1: 761
381      3: 3049
382      0: 383
383   6393: 116939451859...241620750337
384      1: 769
385      8: 98561
386      1: 773
387      2: 1549
388      0: 389
389     11: 796673
390      3: 3121
391      4: 6257
392      3: 3137
393      1: 787
394     14: 6455297
395      5: 12641
396      0: 397
397      4: 6353
398      1: 797
399      2: 1597
400      0: 401

jq

Works with: jq

Works with gojq, the Go implementation of jq.

gojq supports unbounded-precision integer arithmetic but the following algorithm for prime number detection is not up to the stretch tasks.

# Input should be an integer
# No sqrt!
def isPrime:
  . as $n
  | if   ($n < 2)       then false
    elif ($n % 2 == 0)  then $n == 2
    elif ($n % 3 == 0)  then $n == 3
    else 5
    | until( . <= 0;
        if .*. > $n then -1
	elif ($n % . == 0) then 0
        else . + 2
        |  if ($n % . == 0) then 0
           else . + 4
           end
        end)
     | . == -1
     end;

# Emit [m, n*2**m+1] where m is smallest non-negative integer such that n * 2**m + 1 is prime
# WARNING: continues searching ad infinitum ...
def n2m1:
  . as $n
  | first(
      foreach range(0; infinite) as $m (null;
        if . == null then 1 else 2*. end;
        (. * $n + 1)
        | select(isPrime) | [$m, .] ) ) ;

# The task:
"[N,M,Prime]\n------------------",
( range(1;45) | [.] + n2m1 )

Invocation: jq -nrc -f n2m1.jq

Output:
[1,0,2]
[2,0,3]
[3,1,7]
[4,0,5]
[5,1,11]
[6,0,7]
[7,2,29]
[8,1,17]
[9,1,19]
[10,0,11]
[11,1,23]
[12,0,13]
[13,2,53]
[14,1,29]
[15,1,31]
[16,0,17]
[17,3,137]
[18,0,19]
[19,6,1217]
[20,1,41]
[21,1,43]
[22,0,23]
[23,1,47]
[24,2,97]
[25,2,101]
[26,1,53]
[27,2,109]
[28,0,29]
[29,1,59]
[30,0,31]
[31,8,7937]
[32,3,257]
[33,1,67]
[34,2,137]
[35,1,71]
[36,0,37]
[37,2,149]
[38,5,1217]
[39,1,79]
[40,0,41]
[41,1,83]
[42,0,43]
[43,2,173]
[44,1,89]
[45,2,181]
[46,0,47]

Julia

""" Rosetta code task: rosettacode.org/wiki/Primes:_n*2%5Em%2B1 """

using Primes

""" Return true if there is an m such that n * 2**m + 1 is prime """
function n2m1(n)
    for m in big"0":big"10"^300
        isprime(n * big"2"^m + 1) && return true, m
    end
    return false, 0
end

println("  N      M  Prime\n------------------")
for n in 1:400
    tf, m = n2m1(n)
    tf && println(lpad(n, 5), lpad(m, 5), "  ", n * big"2"^m + 1)
end
Output:
  N      M  Prime
------------------
    1    0  2
    2    0  3
    3    1  7
    4    0  5
    5    1  11
    6    0  7
    7    2  29
    8    1  17
    9    1  19
   10    0  11
   11    1  23
   12    0  13
   13    2  53
   14    1  29
   15    1  31
   16    0  17
   17    3  137
   18    0  19
   19    6  1217
   20    1  41
   21    1  43
   22    0  23
   23    1  47
   24    2  97
   25    2  101
   26    1  53
   27    2  109
   28    0  29
   29    1  59
   30    0  31
   31    8  7937
   32    3  257
   33    1  67
   34    2  137
   35    1  71
   36    0  37
   37    2  149
   38    5  1217
   39    1  79
   40    0  41
   41    1  83
   42    0  43
   43    2  173
   44    1  89
   45    2  181
   46    0  47
   47  583  1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
   48    1  97
   49    2  197
   50    1  101
   51    1  103
   52    0  53
   53    1  107
   54    1  109
   55    4  881
   56    1  113
   57    2  229
   58    0  59
   59    5  1889
   60    0  61
   61    4  977
   62    7  7937
   63    1  127
   64    2  257
   65    1  131
   66    0  67
   67    2  269
   68    1  137
   69    1  139
   70    0  71
   71    3  569
   72    0  73
   73    2  293
   74    1  149
   75    1  151
   76    4  1217
   77    3  617
   78    0  79
   79    2  317
   80    3  641
   81    1  163
   82    0  83
   83    1  167
   84    2  337
   85    4  1361
   86    1  173
   87    2  349
   88    0  89
   89    1  179
   90    1  181
   91    8  23297
   92    7  11777
   93    2  373
   94  582  1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
   95    1  191
   96    0  97
   97    2  389
   98    1  197
   99    1  199
  100    0  101
  101    3  809
  102    0  103
  103   16  6750209
  104    5  3329
  105    1  211
  106    0  107
  107    3  857
  108    0  109
  109    6  6977
  110    3  881
  111    1  223
  112    0  113
  113    1  227
  114    1  229
  115    2  461
  116    1  233
  117    3  937
  118    4  1889
  119    1  239
  120    1  241
  121    8  30977
  122    3  977
  123    6  7873
  124    6  7937
  125    1  251
  126    0  127
  127    2  509
  128    1  257
  129    3  1033
  130    0  131
  131    1  263
  132    4  2113
  133    4  2129
  134    1  269
  135    1  271
  136    0  137
  137    3  1097
  138    0  139
  139    2  557
  140    1  281
  141    1  283
  142    2  569
  143   53  1288029493427961857
  144    2  577
  145    6  9281
  146    1  293
  147    8  37633
  148    0  149
  149    3  1193
  150    0  151
  151    4  2417
  152    3  1217
  153    1  307
  154    2  617
  155    1  311
  156    0  157
  157    8  40193
  158    1  317
  159    6  10177
  160    2  641
  161    3  1289
  162    0  163
  163    2  653
  164    9  83969
  165    1  331
  166    0  167
  167    7  21377
  168    1  337
  169    2  677
  170    3  1361
  171    8  43777
  172    0  173
  173    1  347
  174    1  349
  175    2  701
  176    1  353
  177    2  709
  178    0  179
  179    1  359
  180    0  181
  181    4  2897
  182    7  23297
  183    1  367
  184    6  11777
  185    3  1481
  186    1  373
  187    6  11969
  188  581  1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
  189    1  379
  190    0  191
  191    1  383
  192    0  193
  193    2  773
  194    1  389
  195    4  3121
  196    0  197
  197   15  6455297
  198    0  199
  199    2  797
  200    1  401
  201    3  1609
  202    2  809
  203   13  1662977
  204    1  409
  205    2  821
  206   15  6750209
  207    2  829
  208    4  3329
  209    1  419
  210    0  211
  211   20  221249537
  212    3  1697
  213    2  853
  214    2  857
  215    1  431
  216    1  433
  217   66  16011773855979890802689
  218    5  6977
  219    1  439
  220    2  881
  221    1  443
  222    0  223
  223    8  57089
  224    1  449
  225    3  1801
  226    0  227
  227   11  464897
  228    0  229
  229    6  14657
  230    1  461
  231    1  463
  232    0  233
  233    1  467
  234    2  937
  235    2  941
  236    3  1889
  237    4  3793
  238    0  239
  239    1  479
  240    0  241
  241   36  16561393893377
  242    7  30977
  243    1  487
  244    2  977
  245    1  491
  246    5  7873
  247    6  15809
  248    5  7937
  249    1  499
  250    0  251
  251    1  503
  252    2  1009
  253    2  1013
  254    1  509
  255    2  1021
  256    0  257
  257  279  249632952651006185613150855026822179503549278818199928480857894651449200648869292015617
  258    2  1033
  259   38  71193377898497
  260    1  521
  261    1  523
  262    0  263
  263   29  141197049857
  264    3  2113
  265    2  1061
  266    3  2129
  267    2  1069
  268    0  269
  269    3  2153
  270    0  271
  271    4  4337
  272   11  557057
  273    1  547
  274    2  1097
  275    7  35201
  276    0  277
  277    2  1109
  278    1  557
  279    2  1117
  280    0  281
  281    1  563
  282    0  283
  283   30  303868936193
  284    1  569
  285    1  571
  286   52  1288029493427961857
  287    3  2297
  288    1  577
  289   10  295937
  290    5  9281
  291    4  4657
  292    0  293
  293    1  587
  294    7  37633
  295    2  1181
  296    1  593
  297    3  2377
  298    2  1193
  299    1  599
  300    1  601
  301    4  4817
  302    3  2417
  303    1  607
  304    2  1217
  305    3  2441
  306    0  307
  307    2  1229
  308    1  617
  309    1  619
  310    0  311
  311    9  159233
  312    0  313
  313    4  5009
  314    7  40193
  315    1  631
  316    0  317
  317    7  40577
  318    5  10177
  319    2  1277
  320    1  641
  321    1  643
  322    2  1289
  323    1  647
  324    2  1297
  325    2  1301
  326    1  653
  327    3  2617
  328    8  83969
  329    1  659
  330    0  331
  331    4  5297
  332    3  2657
  333    5  10657
  334    6  21377
  335   19  175636481
  336    0  337
  337    4  5393
  338    1  677
  339    3  2713
  340    2  1361
  341    1  683
  342    7  43777
  343    2  1373
  344    3  2753
  345    1  691
  346    0  347
  347    3  2777
  348    0  349
  349   10  357377
  350    1  701
  351   12  1437697
  352    0  353
  353   21  740294657
  354    1  709
  355    6  22721
  356    5  11393
  357    2  1429
  358    0  359
  359    1  719
  360    6  23041
  361   28  96905199617
  362    3  2897
  363    1  727
  364    6  23297
  365    5  11681
  366    0  367
  367   12  1503233
  368    5  11777
  369    1  739
  370    2  1481
  371    1  743
  372    0  373
  373    2  1493
  374    5  11969
  375    1  751
  376  580  1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
  377   11  772097
  378    0  379
  379   14  6209537
  380    1  761
  381    3  3049
  382    0  383
  383 6393  11693945185971565896920916176753769281418376445302724140914106576604960252116205468905429628661873192664799900323401294531072465400997845029722990758855393414014415817179228695517839305455702961095094596926622802342799137107509767542153683280899327558274011281588755909890607960835140712630830933978801393590855371457894042968287926562847826310125559303901351824980311279986492793008248059208985097459095049075732193161126922389950080848742183055141518931962329796357335158955758486061360294773463111842316561192036096585088267052290025273980611139612478214293303564141730470933187279751846912161098280963960686648202780382930927114525552446602357404550468641236474238897222372272898562140228039886991631673186995098587756569010989657598363351856992206826342175536967926902668804937341514786382018872919876784539436965319822540039220122728568129762675989071883516915894567537630751801497223803135172643203770169327233350522822938630733126833423559124391441973547309619943019237705312515304113424366223388373606440335025932390399945086075175009569272136997988977568262327875607690344516747889133920438003737328060362069562108376086129279385800262195985974144460914705464874882401864174074796383557151951711000378565395148939760434428093058777242253682181813425273399277638142811972296863003382484684788329148214958434057306251885787781329925372401240556666727438408378656900945061970219566055969587385482421092779185798692904507774583223151161566406541599486350580593707153172641891804260963429951215526999443852964537303345106153870841180251403751871193132336680841124129779119999935597712685839886558769823834654994044516702436738265181698869580022472787153167463772595005393815295009535991557511340157179280662197799109181549751673455040271529561595718940092424231253150263268513067972937042222806102175350331146290864120703025608712817763221723427454002746818270565050919821097445991953785331131470462682015972815241620750337
  384    1  769
  385    8  98561
  386    1  773
  387    2  1549
  388    0  389
  389   11  796673
  390    3  3121
  391    4  6257
  392    3  3137
  393    1  787
  394   14  6455297
  395    5  12641
  396    0  397
  397    4  6353
  398    1  797
  399    2  1597
  400    0  401

Nim

Task

import std/strformat

func isPrime(n: Natural): bool =
  if n < 2: return false
  if (n and 1) == 0: return n == 2
  if n mod 3 == 0: return n == 3
  var k = 5
  var delta = 2
  while k * k <= n:
    if n mod k == 0: return false
    inc k, delta
    delta = 6 - delta
  result = true

echo " n  m  prime"
for n in 1..45:
  var m = 0
  var term = n
  while true:
    if isPrime(term + 1):
      echo &"{n:2}  {m}  {term + 1:5}"
      break
    inc m
    term *= 2
Output:
 n  m  prime
 1  0      2
 2  0      3
 3  1      7
 4  0      5
 5  1     11
 6  0      7
 7  2     29
 8  1     17
 9  1     19
10  0     11
11  1     23
12  0     13
13  2     53
14  1     29
15  1     31
16  0     17
17  3    137
18  0     19
19  6   1217
20  1     41
21  1     43
22  0     23
23  1     47
24  2     97
25  2    101
26  1     53
27  2    109
28  0     29
29  1     59
30  0     31
31  8   7937
32  3    257
33  1     67
34  2    137
35  1     71
36  0     37
37  2    149
38  5   1217
39  1     79
40  0     41
41  1     83
42  0     43
43  2    173
44  1     89
45  2    181

Stretch tasks

Library: Nim-Integers
import std/strformat
import integers

func compressed(str: string; size: int): string =
  ## Return a compressed value for long strings of digits.
  if str.len <= 2 * size: str
  else: &"{str[0..<size]}...{str[^size..^1]} ({str.len} digits)"

echo "  n     m    prime"
for n in 1..400:
  var m = 0
  var term = newInteger(n)
  while true:
    if isPrime(term + 1):
      echo &"{n:3}  {m:4}    {compressed($(term + 1), 10)}"
      break
    inc m
    term *= 2
Output:
  n     m    prime
  1     0    2
  2     0    3
  3     1    7
  4     0    5
  5     1    11
  6     0    7
  7     2    29
  8     1    17
  9     1    19
 10     0    11
 11     1    23
 12     0    13
 13     2    53
 14     1    29
 15     1    31
 16     0    17
 17     3    137
 18     0    19
 19     6    1217
 20     1    41
 21     1    43
 22     0    23
 23     1    47
 24     2    97
 25     2    101
 26     1    53
 27     2    109
 28     0    29
 29     1    59
 30     0    31
 31     8    7937
 32     3    257
 33     1    67
 34     2    137
 35     1    71
 36     0    37
 37     2    149
 38     5    1217
 39     1    79
 40     0    41
 41     1    83
 42     0    43
 43     2    173
 44     1    89
 45     2    181
 46     0    47
 47   583    1487939695...6574002177 (178 digits)
 48     1    97
 49     2    197
 50     1    101
 51     1    103
 52     0    53
 53     1    107
 54     1    109
 55     4    881
 56     1    113
 57     2    229
 58     0    59
 59     5    1889
 60     0    61
 61     4    977
 62     7    7937
 63     1    127
 64     2    257
 65     1    131
 66     0    67
 67     2    269
 68     1    137
 69     1    139
 70     0    71
 71     3    569
 72     0    73
 73     2    293
 74     1    149
 75     1    151
 76     4    1217
 77     3    617
 78     0    79
 79     2    317
 80     3    641
 81     1    163
 82     0    83
 83     1    167
 84     2    337
 85     4    1361
 86     1    173
 87     2    349
 88     0    89
 89     1    179
 90     1    181
 91     8    23297
 92     7    11777
 93     2    373
 94   582    1487939695...6574002177 (178 digits)
 95     1    191
 96     0    97
 97     2    389
 98     1    197
 99     1    199
100     0    101
101     3    809
102     0    103
103    16    6750209
104     5    3329
105     1    211
106     0    107
107     3    857
108     0    109
109     6    6977
110     3    881
111     1    223
112     0    113
113     1    227
114     1    229
115     2    461
116     1    233
117     3    937
118     4    1889
119     1    239
120     1    241
121     8    30977
122     3    977
123     6    7873
124     6    7937
125     1    251
126     0    127
127     2    509
128     1    257
129     3    1033
130     0    131
131     1    263
132     4    2113
133     4    2129
134     1    269
135     1    271
136     0    137
137     3    1097
138     0    139
139     2    557
140     1    281
141     1    283
142     2    569
143    53    1288029493427961857
144     2    577
145     6    9281
146     1    293
147     8    37633
148     0    149
149     3    1193
150     0    151
151     4    2417
152     3    1217
153     1    307
154     2    617
155     1    311
156     0    157
157     8    40193
158     1    317
159     6    10177
160     2    641
161     3    1289
162     0    163
163     2    653
164     9    83969
165     1    331
166     0    167
167     7    21377
168     1    337
169     2    677
170     3    1361
171     8    43777
172     0    173
173     1    347
174     1    349
175     2    701
176     1    353
177     2    709
178     0    179
179     1    359
180     0    181
181     4    2897
182     7    23297
183     1    367
184     6    11777
185     3    1481
186     1    373
187     6    11969
188   581    1487939695...6574002177 (178 digits)
189     1    379
190     0    191
191     1    383
192     0    193
193     2    773
194     1    389
195     4    3121
196     0    197
197    15    6455297
198     0    199
199     2    797
200     1    401
201     3    1609
202     2    809
203    13    1662977
204     1    409
205     2    821
206    15    6750209
207     2    829
208     4    3329
209     1    419
210     0    211
211    20    221249537
212     3    1697
213     2    853
214     2    857
215     1    431
216     1    433
217    66    1601177385...9890802689 (23 digits)
218     5    6977
219     1    439
220     2    881
221     1    443
222     0    223
223     8    57089
224     1    449
225     3    1801
226     0    227
227    11    464897
228     0    229
229     6    14657
230     1    461
231     1    463
232     0    233
233     1    467
234     2    937
235     2    941
236     3    1889
237     4    3793
238     0    239
239     1    479
240     0    241
241    36    16561393893377
242     7    30977
243     1    487
244     2    977
245     1    491
246     5    7873
247     6    15809
248     5    7937
249     1    499
250     0    251
251     1    503
252     2    1009
253     2    1013
254     1    509
255     2    1021
256     0    257
257   279    2496329526...9292015617 (87 digits)
258     2    1033
259    38    71193377898497
260     1    521
261     1    523
262     0    263
263    29    141197049857
264     3    2113
265     2    1061
266     3    2129
267     2    1069
268     0    269
269     3    2153
270     0    271
271     4    4337
272    11    557057
273     1    547
274     2    1097
275     7    35201
276     0    277
277     2    1109
278     1    557
279     2    1117
280     0    281
281     1    563
282     0    283
283    30    303868936193
284     1    569
285     1    571
286    52    1288029493427961857
287     3    2297
288     1    577
289    10    295937
290     5    9281
291     4    4657
292     0    293
293     1    587
294     7    37633
295     2    1181
296     1    593
297     3    2377
298     2    1193
299     1    599
300     1    601
301     4    4817
302     3    2417
303     1    607
304     2    1217
305     3    2441
306     0    307
307     2    1229
308     1    617
309     1    619
310     0    311
311     9    159233
312     0    313
313     4    5009
314     7    40193
315     1    631
316     0    317
317     7    40577
318     5    10177
319     2    1277
320     1    641
321     1    643
322     2    1289
323     1    647
324     2    1297
325     2    1301
326     1    653
327     3    2617
328     8    83969
329     1    659
330     0    331
331     4    5297
332     3    2657
333     5    10657
334     6    21377
335    19    175636481
336     0    337
337     4    5393
338     1    677
339     3    2713
340     2    1361
341     1    683
342     7    43777
343     2    1373
344     3    2753
345     1    691
346     0    347
347     3    2777
348     0    349
349    10    357377
350     1    701
351    12    1437697
352     0    353
353    21    740294657
354     1    709
355     6    22721
356     5    11393
357     2    1429
358     0    359
359     1    719
360     6    23041
361    28    96905199617
362     3    2897
363     1    727
364     6    23297
365     5    11681
366     0    367
367    12    1503233
368     5    11777
369     1    739
370     2    1481
371     1    743
372     0    373
373     2    1493
374     5    11969
375     1    751
376   580    1487939695...6574002177 (178 digits)
377    11    772097
378     0    379
379    14    6209537
380     1    761
381     3    3049
382     0    383
383  6393    1169394518...1620750337 (1928 digits)
384     1    769
385     8    98561
386     1    773
387     2    1549
388     0    389
389    11    796673
390     3    3121
391     4    6257
392     3    3137
393     1    787
394    14    6455297
395     5    12641
396     0    397
397     4    6353
398     1    797
399     2    1597
400     0    401

Oberon-07

(* find primes of the form 1+n*2^m                                           *)
(* where m is the lowest integer >= 0 such that 1+n*2^m is prime             *)
MODULE primesNx2ToMPlus1;
IMPORT
    Math, Out;

CONST
    MaxM     = 22;                             (* maximum m we will consider *)
    MaxPrime = 10000;                                          (* sieve size *)

VAR prime :ARRAY MaxPrime + 1 OF BOOLEAN;

PROCEDURE Sieve;                             (* Sieve the primes to MaxPrime *)
    VAR i, ii, pr    :INTEGER;
    BEGIN
        prime[ 0 ] := FALSE; prime[ 1 ] := FALSE; prime[ 2 ] := TRUE;
        FOR i := 4 TO MaxPrime BY 2 DO prime[ i ] := FALSE END;
        FOR i := 3 TO MaxPrime BY 2 DO prime[ i ] := TRUE  END;
        FOR i := 3 TO FLOOR( Math.sqrt( FLT( MaxPrime ) ) ) BY 2 DO
            IF prime[ i ] THEN
                ii := i + i;
                pr := i * i;
                WHILE pr <= MaxPrime DO
                    prime[ pr ] := FALSE;
                    pr := pr + ii
                END
            END
        END
    END Sieve;

PROCEDURE ShowPrimes;                           (* find the n*2^m + 1 primes *)
    VAR n, m, nx2ToM, p    :INTEGER;
        found              :BOOLEAN;
    BEGIN
        FOR n := 1 TO 45 DO
            m      := 0;
            nx2ToM := n;
            p      := 0;
            found  := FALSE;
            WHILE ~ found & ( m <= MaxM ) DO
                p := nx2ToM + 1;
                found := prime[ p ];
                IF ~ found THEN
                    nx2ToM := nx2ToM + nx2ToM;
                    m      := m + 1
                END
            END;
            Out.Int( n, 3 );
            IF ~ found THEN
                Out.String( " not found)" )
            ELSE
                Out.Int( m, 4 );Out.String( ": " );Out.Int( p, 0 )
            END;
            Out.Ln
        END
    END ShowPrimes;

BEGIN
    Sieve;
    ShowPrimes;
END primesNx2ToMPlus1.
Output:
  1   0: 2
  2   0: 3
  3   1: 7
  4   0: 5
  5   1: 11
  6   0: 7
  7   2: 29
  8   1: 17
  9   1: 19
 10   0: 11
 11   1: 23
 12   0: 13
 13   2: 53
 14   1: 29
 15   1: 31
 16   0: 17
 17   3: 137
 18   0: 19
 19   6: 1217
 20   1: 41
 21   1: 43
 22   0: 23
 23   1: 47
 24   2: 97
 25   2: 101
 26   1: 53
 27   2: 109
 28   0: 29
 29   1: 59
 30   0: 31
 31   8: 7937
 32   3: 257
 33   1: 67
 34   2: 137
 35   1: 71
 36   0: 37
 37   2: 149
 38   5: 1217
 39   1: 79
 40   0: 41
 41   1: 83
 42   0: 43
 43   2: 173
 44   1: 89
 45   2: 181

PARI/GP

Translation of: Julia
/* Check if there is an m such that n * 2^m + 1 is prime */
n2m1(n) = {
   for(m = 0, 10^300,
       if(isprime(n * 2^m + 1), return([1, m]));
   );
   return([0, 0]);
}


{
    print("  N      M  Prime\n------------------");
    for(n = 1, 400,
        result = n2m1(n);
        if(result[1],
            print(Str(n) " " Str(result[2]) "  " n * 2^result[2] + 1);
        );
    );
}
Output:
  N      M  Prime
------------------
1 0  2
2 0  3
3 1  7
4 0  5
5 1  11
6 0  7
7 2  29
8 1  17
9 1  19
10 0  11
11 1  23
12 0  13
13 2  53
14 1  29
15 1  31
16 0  17
17 3  137
18 0  19
19 6  1217
20 1  41
21 1  43
22 0  23
23 1  47
24 2  97
25 2  101
26 1  53
27 2  109
28 0  29
29 1  59
30 0  31
31 8  7937
32 3  257
33 1  67
34 2  137
35 1  71
36 0  37
37 2  149
38 5  1217
39 1  79
40 0  41
41 1  83
42 0  43
43 2  173
44 1  89
45 2  181
46 0  47
47 583  1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
48 1  97
49 2  197
50 1  101
51 1  103
52 0  53
53 1  107
54 1  109
55 4  881
56 1  113
57 2  229
58 0  59
59 5  1889
60 0  61
61 4  977
62 7  7937
63 1  127
64 2  257
65 1  131
66 0  67
67 2  269
68 1  137
69 1  139
70 0  71
71 3  569
72 0  73
73 2  293
74 1  149
75 1  151
76 4  1217
77 3  617
78 0  79
79 2  317
80 3  641
81 1  163
82 0  83
83 1  167
84 2  337
85 4  1361
86 1  173
87 2  349
88 0  89
89 1  179
90 1  181
91 8  23297
92 7  11777
93 2  373
94 582  1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
95 1  191
96 0  97
97 2  389
98 1  197
99 1  199
100 0  101
101 3  809
102 0  103
103 16  6750209
104 5  3329
105 1  211
106 0  107
107 3  857
108 0  109
109 6  6977
110 3  881
111 1  223
112 0  113
113 1  227
114 1  229
115 2  461
116 1  233
117 3  937
118 4  1889
119 1  239
120 1  241
121 8  30977
122 3  977
123 6  7873
124 6  7937
125 1  251
126 0  127
127 2  509
128 1  257
129 3  1033
130 0  131
131 1  263
132 4  2113
133 4  2129
134 1  269
135 1  271
136 0  137
137 3  1097
138 0  139
139 2  557
140 1  281
141 1  283
142 2  569
143 53  1288029493427961857
144 2  577
145 6  9281
146 1  293
147 8  37633
148 0  149
149 3  1193
150 0  151
151 4  2417
152 3  1217
153 1  307
154 2  617
155 1  311
156 0  157
157 8  40193
158 1  317
159 6  10177
160 2  641
161 3  1289
162 0  163
163 2  653
164 9  83969
165 1  331
166 0  167
167 7  21377
168 1  337
169 2  677
170 3  1361
171 8  43777
172 0  173
173 1  347
174 1  349
175 2  701
176 1  353
177 2  709
178 0  179
179 1  359
180 0  181
181 4  2897
182 7  23297
183 1  367
184 6  11777
185 3  1481
186 1  373
187 6  11969
188 581  1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
189 1  379
190 0  191
191 1  383
192 0  193
193 2  773
194 1  389
195 4  3121
196 0  197
197 15  6455297
198 0  199
199 2  797
200 1  401
201 3  1609
202 2  809
203 13  1662977
204 1  409
205 2  821
206 15  6750209
207 2  829
208 4  3329
209 1  419
210 0  211
211 20  221249537
212 3  1697
213 2  853
214 2  857
215 1  431
216 1  433
217 66  16011773855979890802689
218 5  6977
219 1  439
220 2  881
221 1  443
222 0  223
223 8  57089
224 1  449
225 3  1801
226 0  227
227 11  464897
228 0  229
229 6  14657
230 1  461
231 1  463
232 0  233
233 1  467
234 2  937
235 2  941
236 3  1889
237 4  3793
238 0  239
239 1  479
240 0  241
241 36  16561393893377
242 7  30977
243 1  487
244 2  977
245 1  491
246 5  7873
247 6  15809
248 5  7937
249 1  499
250 0  251
251 1  503
252 2  1009
253 2  1013
254 1  509
255 2  1021
256 0  257
257 279  249632952651006185613150855026822179503549278818199928480857894651449200648869292015617
258 2  1033
259 38  71193377898497
260 1  521
261 1  523
262 0  263
263 29  141197049857
264 3  2113
265 2  1061
266 3  2129
267 2  1069
268 0  269
269 3  2153
270 0  271
271 4  4337
272 11  557057
273 1  547
274 2  1097
275 7  35201
276 0  277
277 2  1109
278 1  557
279 2  1117
280 0  281
281 1  563
282 0  283
283 30  303868936193
284 1  569
285 1  571
286 52  1288029493427961857
287 3  2297
288 1  577
289 10  295937
290 5  9281
291 4  4657
292 0  293
293 1  587
294 7  37633
295 2  1181
296 1  593
297 3  2377
298 2  1193
299 1  599
300 1  601
301 4  4817
302 3  2417
303 1  607
304 2  1217
305 3  2441
306 0  307
307 2  1229
308 1  617
309 1  619
310 0  311
311 9  159233
312 0  313
313 4  5009
314 7  40193
315 1  631
316 0  317
317 7  40577
318 5  10177
319 2  1277
320 1  641
321 1  643
322 2  1289
323 1  647
324 2  1297
325 2  1301
326 1  653
327 3  2617
328 8  83969
329 1  659
330 0  331
331 4  5297
332 3  2657
333 5  10657
334 6  21377
335 19  175636481
336 0  337
337 4  5393
338 1  677
339 3  2713
340 2  1361
341 1  683
342 7  43777
343 2  1373
344 3  2753
345 1  691
346 0  347
347 3  2777
348 0  349
349 10  357377
350 1  701
351 12  1437697
352 0  353
353 21  740294657
354 1  709
355 6  22721
356 5  11393
357 2  1429
358 0  359
359 1  719
360 6  23041
361 28  96905199617
362 3  2897
363 1  727
364 6  23297
365 5  11681
366 0  367
367 12  1503233
368 5  11777
369 1  739
370 2  1481
371 1  743
372 0  373
373 2  1493
374 5  11969
375 1  751
376 580  1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
377 11  772097
378 0  379
379 14  6209537
380 1  761
381 3  3049
382 0  383
383 6393  11693945185971565896920916176753769281418376445302724140914106576604960252116205468905429628661873192664799900323401294531072465400997845029722990758855393414014415817179228695517839305455702961095094596926622802342799137107509767542153683280899327558274011281588755909890607960835140712630830933978801393590855371457894042968287926562847826310125559303901351824980311279986492793008248059208985097459095049075732193161126922389950080848742183055141518931962329796357335158955758486061360294773463111842316561192036096585088267052290025273980611139612478214293303564141730470933187279751846912161098280963960686648202780382930927114525552446602357404550468641236474238897222372272898562140228039886991631673186995098587756569010989657598363351856992206826342175536967926902668804937341514786382018872919876784539436965319822540039220122728568129762675989071883516915894567537630751801497223803135172643203770169327233350522822938630733126833423559124391441973547309619943019237705312515304113424366223388373606440335025932390399945086075175009569272136997988977568262327875607690344516747889133920438003737328060362069562108376086129279385800262195985974144460914705464874882401864174074796383557151951711000378565395148939760434428093058777242253682181813425273399277638142811972296863003382484684788329148214958434057306251885787781329925372401240556666727438408378656900945061970219566055969587385482421092779185798692904507774583223151161566406541599486350580593707153172641891804260963429951215526999443852964537303345106153870841180251403751871193132336680841124129779119999935597712685839886558769823834654994044516702436738265181698869580022472787153167463772595005393815295009535991557511340157179280662197799109181549751673455040271529561595718940092424231253150263268513067972937042222806102175350331146290864120703025608712817763221723427454002746818270565050919821097445991953785331131470462682015972815241620750337
384 1  769
385 8  98561
386 1  773
387 2  1549
388 0  389
389 11  796673
390 3  3121
391 4  6257
392 3  3137
393 1  787
394 14  6455297
395 5  12641
396 0  397
397 4  6353
398 1  797
399 2  1597
400 0  401

Perl

Library: ntheory
use v5.36;
use bigint;
use ntheory 'is_prime';

for my $n (1..400) {
    for (my $m=0 ; ; $m += 1) {
        if (is_prime(my $p = $n * 2**$m + 1)) { printf "%3d %4d: %s\n",$n,$m,$p; last }
    }
}
Output:
  1    0: 2
  2    0: 3
  3    1: 7
  4    0: 5
  5    1: 11
  6    0: 7
  7    2: 29
  8    1: 17
  9    1: 19
 10    0: 11
 11    1: 23
 12    0: 13
 13    2: 53
 14    1: 29
 15    1: 31
 16    0: 17
 17    3: 137
 18    0: 19
 19    6: 1217
 20    1: 41
 21    1: 43
 22    0: 23
 23    1: 47
 24    2: 97
 25    2: 101
 26    1: 53
 27    2: 109
 28    0: 29
 29    1: 59
 30    0: 31
 31    8: 7937
 32    3: 257
 33    1: 67
 34    2: 137
 35    1: 71
 36    0: 37
 37    2: 149
 38    5: 1217
 39    1: 79
 40    0: 41
 41    1: 83
 42    0: 43
 43    2: 173
 44    1: 89
 45    2: 181
 46    0: 47
 47  583: 1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
 48    1: 97
 49    2: 197
 50    1: 101
 51    1: 103
 52    0: 53
 53    1: 107
 54    1: 109
 55    4: 881
 56    1: 113
 57    2: 229
 58    0: 59
 59    5: 1889
 60    0: 61
 61    4: 977
 62    7: 7937
 63    1: 127
 64    2: 257
 65    1: 131
 66    0: 67
 67    2: 269
 68    1: 137
 69    1: 139
 70    0: 71
 71    3: 569
 72    0: 73
 73    2: 293
 74    1: 149
 75    1: 151
 76    4: 1217
 77    3: 617
 78    0: 79
 79    2: 317
 80    3: 641
 81    1: 163
 82    0: 83
 83    1: 167
 84    2: 337
 85    4: 1361
 86    1: 173
 87    2: 349
 88    0: 89
 89    1: 179
 90    1: 181
 91    8: 23297
 92    7: 11777
 93    2: 373
 94  582: 1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
 95    1: 191
 96    0: 97
 97    2: 389
 98    1: 197
 99    1: 199
100    0: 101
101    3: 809
102    0: 103
103   16: 6750209
104    5: 3329
105    1: 211
106    0: 107
107    3: 857
108    0: 109
109    6: 6977
110    3: 881
111    1: 223
112    0: 113
113    1: 227
114    1: 229
115    2: 461
116    1: 233
117    3: 937
118    4: 1889
119    1: 239
120    1: 241
121    8: 30977
122    3: 977
123    6: 7873
124    6: 7937
125    1: 251
126    0: 127
127    2: 509
128    1: 257
129    3: 1033
130    0: 131
131    1: 263
132    4: 2113
133    4: 2129
134    1: 269
135    1: 271
136    0: 137
137    3: 1097
138    0: 139
139    2: 557
140    1: 281
141    1: 283
142    2: 569
143   53: 1288029493427961857
144    2: 577
145    6: 9281
146    1: 293
147    8: 37633
148    0: 149
149    3: 1193
150    0: 151
151    4: 2417
152    3: 1217
153    1: 307
154    2: 617
155    1: 311
156    0: 157
157    8: 40193
158    1: 317
159    6: 10177
160    2: 641
161    3: 1289
162    0: 163
163    2: 653
164    9: 83969
165    1: 331
166    0: 167
167    7: 21377
168    1: 337
169    2: 677
170    3: 1361
171    8: 43777
172    0: 173
173    1: 347
174    1: 349
175    2: 701
176    1: 353
177    2: 709
178    0: 179
179    1: 359
180    0: 181
181    4: 2897
182    7: 23297
183    1: 367
184    6: 11777
185    3: 1481
186    1: 373
187    6: 11969
188  581: 1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
189    1: 379
190    0: 191
191    1: 383
192    0: 193
193    2: 773
194    1: 389
195    4: 3121
196    0: 197
197   15: 6455297
198    0: 199
199    2: 797
200    1: 401
201    3: 1609
202    2: 809
203   13: 1662977
204    1: 409
205    2: 821
206   15: 6750209
207    2: 829
208    4: 3329
209    1: 419
210    0: 211
211   20: 221249537
212    3: 1697
213    2: 853
214    2: 857
215    1: 431
216    1: 433
217   66: 16011773855979890802689
218    5: 6977
219    1: 439
220    2: 881
221    1: 443
222    0: 223
223    8: 57089
224    1: 449
225    3: 1801
226    0: 227
227   11: 464897
228    0: 229
229    6: 14657
230    1: 461
231    1: 463
232    0: 233
233    1: 467
234    2: 937
235    2: 941
236    3: 1889
237    4: 3793
238    0: 239
239    1: 479
240    0: 241
241   36: 16561393893377
242    7: 30977
243    1: 487
244    2: 977
245    1: 491
246    5: 7873
247    6: 15809
248    5: 7937
249    1: 499
250    0: 251
251    1: 503
252    2: 1009
253    2: 1013
254    1: 509
255    2: 1021
256    0: 257
257  279: 249632952651006185613150855026822179503549278818199928480857894651449200648869292015617
258    2: 1033
259   38: 71193377898497
260    1: 521
261    1: 523
262    0: 263
263   29: 141197049857
264    3: 2113
265    2: 1061
266    3: 2129
267    2: 1069
268    0: 269
269    3: 2153
270    0: 271
271    4: 4337
272   11: 557057
273    1: 547
274    2: 1097
275    7: 35201
276    0: 277
277    2: 1109
278    1: 557
279    2: 1117
280    0: 281
281    1: 563
282    0: 283
283   30: 303868936193
284    1: 569
285    1: 571
286   52: 1288029493427961857
287    3: 2297
288    1: 577
289   10: 295937
290    5: 9281
291    4: 4657
292    0: 293
293    1: 587
294    7: 37633
295    2: 1181
296    1: 593
297    3: 2377
298    2: 1193
299    1: 599
300    1: 601
301    4: 4817
302    3: 2417
303    1: 607
304    2: 1217
305    3: 2441
306    0: 307
307    2: 1229
308    1: 617
309    1: 619
310    0: 311
311    9: 159233
312    0: 313
313    4: 5009
314    7: 40193
315    1: 631
316    0: 317
317    7: 40577
318    5: 10177
319    2: 1277
320    1: 641
321    1: 643
322    2: 1289
323    1: 647
324    2: 1297
325    2: 1301
326    1: 653
327    3: 2617
328    8: 83969
329    1: 659
330    0: 331
331    4: 5297
332    3: 2657
333    5: 10657
334    6: 21377
335   19: 175636481
336    0: 337
337    4: 5393
338    1: 677
339    3: 2713
340    2: 1361
341    1: 683
342    7: 43777
343    2: 1373
344    3: 2753
345    1: 691
346    0: 347
347    3: 2777
348    0: 349
349   10: 357377
350    1: 701
351   12: 1437697
352    0: 353
353   21: 740294657
354    1: 709
355    6: 22721
356    5: 11393
357    2: 1429
358    0: 359
359    1: 719
360    6: 23041
361   28: 96905199617
362    3: 2897
363    1: 727
364    6: 23297
365    5: 11681
366    0: 367
367   12: 1503233
368    5: 11777
369    1: 739
370    2: 1481
371    1: 743
372    0: 373
373    2: 1493
374    5: 11969
375    1: 751
376  580: 1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
377   11: 772097
378    0: 379
379   14: 6209537
380    1: 761
381    3: 3049
382    0: 383
383 6393: 11693945185971565896920916176753769281418376445302724140914106576604960252116205468905429628661873192664799900323401294531072465400997845029722990758855393414014415817179228695517839305455702961095094596926622802342799137107509767542153683280899327558274011281588755909890607960835140712630830933978801393590855371457894042968287926562847826310125559303901351824980311279986492793008248059208985097459095049075732193161126922389950080848742183055141518931962329796357335158955758486061360294773463111842316561192036096585088267052290025273980611139612478214293303564141730470933187279751846912161098280963960686648202780382930927114525552446602357404550468641236474238897222372272898562140228039886991631673186995098587756569010989657598363351856992206826342175536967926902668804937341514786382018872919876784539436965319822540039220122728568129762675989071883516915894567537630751801497223803135172643203770169327233350522822938630733126833423559124391441973547309619943019237705312515304113424366223388373606440335025932390399945086075175009569272136997988977568262327875607690344516747889133920438003737328060362069562108376086129279385800262195985974144460914705464874882401864174074796383557151951711000378565395148939760434428093058777242253682181813425273399277638142811972296863003382484684788329148214958434057306251885787781329925372401240556666727438408378656900945061970219566055969587385482421092779185798692904507774583223151161566406541599486350580593707153172641891804260963429951215526999443852964537303345106153870841180251403751871193132336680841124129779119999935597712685839886558769823834654994044516702436738265181698869580022472787153167463772595005393815295009535991557511340157179280662197799109181549751673455040271529561595718940092424231253150263268513067972937042222806102175350331146290864120703025608712817763221723427454002746818270565050919821097445991953785331131470462682015972815241620750337
384    1: 769
385    8: 98561
386    1: 773
387    2: 1549
388    0: 389
389   11: 796673
390    3: 3121
391    4: 6257
392    3: 3137
393    1: 787
394   14: 6455297
395    5: 12641
396    0: 397
397    4: 6353
398    1: 797
399    2: 1597
400    0: 401

PL/M

Works with: 8080 PL/M Compiler

... under CP/M (or an emulator)

Interestingly, the primes up to m = 45 all have a single digit m and will all fit in 16 bits, which is handy as the 8080 PL/M compiler doesn't support integers larger than unsigned 16-bit.

100H: /* FIND PRIMES OF THE FORM N * 2**M + 1                                */

   DECLARE FALSE LITERALLY '0';
   DECLARE TRUE  LITERALLY '0FFH';

   /* CP/M SYSTEM CALL AND I/O ROUTINES                                      */
   BDOS:      PROCEDURE( FN, ARG ); DECLARE FN BYTE, ARG ADDRESS; GOTO 5; END;
   PR$CHAR:   PROCEDURE( C ); DECLARE C BYTE;    CALL BDOS( 2, C );  END;
   PR$STRING: PROCEDURE( S ); DECLARE S ADDRESS; CALL BDOS( 9, S );  END;
   PR$NL:     PROCEDURE;   CALL PR$CHAR( 0DH ); CALL PR$CHAR( 0AH ); END;
   PR$NUMBER: PROCEDURE( N ); /* PRINTS A NUMBER IN THE MINIMUN FIELD WIDTH  */
      DECLARE N ADDRESS;
      DECLARE V ADDRESS, N$STR ( 6 )BYTE, W BYTE;
      V = N;
      W = LAST( N$STR );
      N$STR( W ) = '$';
      N$STR( W := W - 1 ) = '0' + ( V MOD 10 );
      DO WHILE( ( V := V / 10 ) > 0 );
         N$STR( W := W - 1 ) = '0' + ( V MOD 10 );
      END;
      CALL PR$STRING( .N$STR( W ) );
   END PR$NUMBER;
   /* END SYSTEM CALL AND I/O ROUTINES                                       */

   /* SIEVE THE PRIMES TO 8000                                               */
   DECLARE PRIME ( 8001 )BYTE;
   DO;
      DECLARE ( I, S ) ADDRESS;
      PRIME( 0 ),  PRIME( 1 ) = FALSE;
      PRIME( 2 ) = TRUE;
      DO I = 3 TO LAST( PRIME ) BY 2; PRIME( I ) = TRUE;  END;
      DO I = 4 TO LAST( PRIME ) BY 2; PRIME( I ) = FALSE; END;
      DO I = 3 TO LAST( PRIME ) / 2 BY 2;
         IF PRIME( I ) THEN DO;
            DO S = I * I TO LAST( PRIME ) BY I + I; PRIME( S ) = FALSE; END;
         END;
      END;
   END;

   DECLARE MAX$M LITERALLY '14'; /* MAXIMUM M WE WILL CONSIDER               */

   /* FIND THE PRIMES                                                        */

   DECLARE ( N, M, P, TWO$TO$M ) ADDRESS;
   DECLARE NOT$FOUND             BYTE;

   DO N = 1 TO 45;
      M         = 0;
      TWO$TO$M  = 1;
      P         = 0;
      NOT$FOUND = 1;
      DO WHILE M <= MAX$M
           AND ( NOT$FOUND := NOT PRIME( P := ( N * TWO$TO$M ) + 1 ) );
         TWO$TO$M = TWO$TO$M + TWO$TO$M;
         M        = M + 1;
      END;
      CALL PR$CHAR( '(' );
      IF N < 10 THEN CALL PR$CHAR( ' ' );
      CALL PR$NUMBER( N );
      IF NOT$FOUND THEN DO;
         CALL PR$STRING( .' NOT FOUND$' );
         END;
      ELSE DO;
         CALL PR$CHAR( ' ' );
         CALL PR$NUMBER( M );
         CALL PR$CHAR( ':' );
         CALL PR$CHAR( ' ' );
         IF P <   10 THEN CALL PR$CHAR( ' ' );
         IF P <  100 THEN CALL PR$CHAR( ' ' );
         IF P < 1000 THEN CALL PR$CHAR( ' ' );
         CALL PR$NUMBER( P );
         CALL PR$CHAR( ' ' );
         CALL PR$CHAR( ' ' );
      END;
      CALL PR$CHAR( ')' );
      IF N MOD 5 = 0 THEN CALL PR$NL;
   END;

EOF
Output:
( 1 0:    2  )( 2 0:    3  )( 3 1:    7  )( 4 0:    5  )( 5 1:   11  )
( 6 0:    7  )( 7 2:   29  )( 8 1:   17  )( 9 1:   19  )(10 0:   11  )
(11 1:   23  )(12 0:   13  )(13 2:   53  )(14 1:   29  )(15 1:   31  )
(16 0:   17  )(17 3:  137  )(18 0:   19  )(19 6: 1217  )(20 1:   41  )
(21 1:   43  )(22 0:   23  )(23 1:   47  )(24 2:   97  )(25 2:  101  )
(26 1:   53  )(27 2:  109  )(28 0:   29  )(29 1:   59  )(30 0:   31  )
(31 8: 7937  )(32 3:  257  )(33 1:   67  )(34 2:  137  )(35 1:   71  )
(36 0:   37  )(37 2:  149  )(38 5: 1217  )(39 1:   79  )(40 0:   41  )
(41 1:   83  )(42 0:   43  )(43 2:  173  )(44 1:   89  )(45 2:  181  )

Phix

Translation of: Wren
with javascript_semantics
include mpfr.e
printf(1,"  N     M    Prime\n------------------\n")
mpz p = mpz_init()
for n=1 to 400 do
    integer m = 0
    while true do
        mpz_set_si(p,n)
        mpz_mul_2exp(p,p,m)
        mpz_add_si(p,p,1)
        if mpz_prime(p) then
            printf(1,"%3d  %4d    %s\n", {n,m,mpz_get_short_str(p)})
            exit
        end if
        m += 1
    end while
end for

Output same as Wren (plus a few not particularly helpful digit counts).

Python

Library: gmpy2
Translation of: Nim
 # primesn2m1.py by Xing216
from gmpy2 import is_prime, mpz

print("  n  m  prime")
for n in range(1, 401):
    m = 0
    term = mpz(n)
    while True:
        if is_prime(term + 1):
            print(f"{n:3d}  {m}  {term + 1:5d}")
            break
        m += 1
        term *= 2
}
Output:
  n  m  prime
  1  0      2
  2  0      3
  3  1      7
  4  0      5
  5  1     11
  6  0      7
  7  2     29
  8  1     17
  9  1     19
 10  0     11
 11  1     23
 12  0     13
 13  2     53
 14  1     29
 15  1     31
 16  0     17
 17  3    137
 18  0     19
 19  6   1217
 20  1     41
 21  1     43
 22  0     23
 23  1     47
 24  2     97
 25  2    101
 26  1     53
 27  2    109
 28  0     29
 29  1     59
 30  0     31
 31  8   7937
 32  3    257
 33  1     67
 34  2    137
 35  1     71
 36  0     37
 37  2    149
 38  5   1217
 39  1     79
 40  0     41
 41  1     83
 42  0     43
 43  2    173
 44  1     89
 45  2    181
 46  0     47
 47  583  1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177    
 48  1     97
 49  2    197
 50  1    101
 51  1    103
 52  0     53
 53  1    107
 54  1    109
 55  4    881
 56  1    113
 57  2    229
 58  0     59
 59  5   1889
 60  0     61
 61  4    977
 62  7   7937
 63  1    127
 64  2    257
 65  1    131
 66  0     67
 67  2    269
 68  1    137
 69  1    139
 70  0     71
 71  3    569
 72  0     73
 73  2    293
 74  1    149
 75  1    151
 76  4   1217
 77  3    617
 78  0     79
 79  2    317
 80  3    641
 81  1    163
 82  0     83
 83  1    167
 84  2    337
 85  4   1361
 86  1    173
 87  2    349
 88  0     89
 89  1    179
 90  1    181
 91  8  23297
 92  7  11777
 93  2    373
 94  582  1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177    
 95  1    191
 96  0     97
 97  2    389
 98  1    197
 99  1    199
100  0    101
101  3    809
102  0    103
103  16  6750209
104  5   3329
105  1    211
106  0    107
107  3    857
108  0    109
109  6   6977
110  3    881
111  1    223
112  0    113
113  1    227
114  1    229
115  2    461
116  1    233
117  3    937
118  4   1889
119  1    239
120  1    241
121  8  30977
122  3    977
123  6   7873
124  6   7937
125  1    251
126  0    127
127  2    509
128  1    257
129  3   1033
130  0    131
131  1    263
132  4   2113
133  4   2129
134  1    269
135  1    271
136  0    137
137  3   1097
138  0    139
139  2    557
140  1    281
141  1    283
142  2    569
143  53  1288029493427961857
144  2    577
145  6   9281
146  1    293
147  8  37633
148  0    149
149  3   1193
150  0    151
151  4   2417
152  3   1217
153  1    307
154  2    617
155  1    311
156  0    157
157  8  40193
158  1    317
159  6  10177
160  2    641
161  3   1289
162  0    163
163  2    653
164  9  83969
165  1    331
166  0    167
167  7  21377
168  1    337
169  2    677
170  3   1361
171  8  43777
172  0    173
173  1    347
174  1    349
175  2    701
176  1    353
177  2    709
178  0    179
179  1    359
180  0    181
181  4   2897
182  7  23297
183  1    367
184  6  11777
185  3   1481
186  1    373
187  6  11969
188  581  1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177    
189  1    379
190  0    191
191  1    383
192  0    193
193  2    773
194  1    389
195  4   3121
196  0    197
197  15  6455297
198  0    199
199  2    797
200  1    401
201  3   1609
202  2    809
203  13  1662977
204  1    409
205  2    821
206  15  6750209
207  2    829
208  4   3329
209  1    419
210  0    211
211  20  221249537
212  3   1697
213  2    853
214  2    857
215  1    431
216  1    433
217  66  16011773855979890802689
218  5   6977
219  1    439
220  2    881
221  1    443
222  0    223
223  8  57089
224  1    449
225  3   1801
226  0    227
227  11  464897
228  0    229
229  6  14657
230  1    461
231  1    463
232  0    233
233  1    467
234  2    937
235  2    941
236  3   1889
237  4   3793
238  0    239
239  1    479
240  0    241
241  36  16561393893377
242  7  30977
243  1    487
244  2    977
245  1    491
246  5   7873
247  6  15809
248  5   7937
249  1    499
250  0    251
251  1    503
252  2   1009
253  2   1013
254  1    509
255  2   1021
256  0    257
257  279  249632952651006185613150855026822179503549278818199928480857894651449200648869292015617
258  2   1033
259  38  71193377898497
260  1    521
261  1    523
262  0    263
263  29  141197049857
264  3   2113
265  2   1061
266  3   2129
267  2   1069
268  0    269
269  3   2153
270  0    271
271  4   4337
272  11  557057
273  1    547
274  2   1097
275  7  35201
276  0    277
277  2   1109
278  1    557
279  2   1117
280  0    281
281  1    563
282  0    283
283  30  303868936193
284  1    569
285  1    571
286  52  1288029493427961857
287  3   2297
288  1    577
289  10  295937
290  5   9281
291  4   4657
292  0    293
293  1    587
294  7  37633
295  2   1181
296  1    593
297  3   2377
298  2   1193
299  1    599
300  1    601
301  4   4817
302  3   2417
303  1    607
304  2   1217
305  3   2441
306  0    307
307  2   1229
308  1    617
309  1    619
310  0    311
311  9  159233
312  0    313
313  4   5009
314  7  40193
315  1    631
316  0    317
317  7  40577
318  5  10177
319  2   1277
320  1    641
321  1    643
322  2   1289
323  1    647
324  2   1297
325  2   1301
326  1    653
327  3   2617
328  8  83969
329  1    659
330  0    331
331  4   5297
332  3   2657
333  5  10657
334  6  21377
335  19  175636481
336  0    337
337  4   5393
338  1    677
339  3   2713
340  2   1361
341  1    683
342  7  43777
343  2   1373
344  3   2753
345  1    691
346  0    347
347  3   2777
348  0    349
349  10  357377
350  1    701
351  12  1437697
352  0    353
353  21  740294657
354  1    709
355  6  22721
356  5  11393
357  2   1429
358  0    359
359  1    719
360  6  23041
361  28  96905199617
362  3   2897
363  1    727
364  6  23297
365  5  11681
366  0    367
367  12  1503233
368  5  11777
369  1    739
370  2   1481
371  1    743
372  0    373
373  2   1493
374  5  11969
375  1    751
376  580  1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
377  11  772097
378  0    379
379  14  6209537
380  1    761
381  3   3049
382  0    383
383  6393  11693945185971565896920916176753769281418376445302724140914106576604960252116205468905429628661873192664799900323401294531072465400997845029722990758855393414014415817179228695517839305455702961095094596926622802342799137107509767542153683280899327558274011281588755909890607960835140712630830933978801393590855371457894042968287926562847826310125559303901351824980311279986492793008248059208985097459095049075732193161126922389950080848742183055141518931962329796357335158955758486061360294773463111842316561192036096585088267052290025273980611139612478214293303564141730470933187279751846912161098280963960686648202780382930927114525552446602357404550468641236474238897222372272898562140228039886991631673186995098587756569010989657598363351856992206826342175536967926902668804937341514786382018872919876784539436965319822540039220122728568129762675989071883516915894567537630751801497223803135172643203770169327233350522822938630733126833423559124391441973547309619943019237705312515304113424366223388373606440335025932390399945086075175009569272136997988977568262327875607690344516747889133920438003737328060362069562108376086129279385800262195985974144460914705464874882401864174074796383557151951711000378565395148939760434428093058777242253682181813425273399277638142811972296863003382484684788329148214958434057306251885787781329925372401240556666727438408378656900945061970219566055969587385482421092779185798692904507774583223151161566406541599486350580593707153172641891804260963429951215526999443852964537303345106153870841180251403751871193132336680841124129779119999935597712685839886558769823834654994044516702436738265181698869580022472787153167463772595005393815295009535991557511340157179280662197799109181549751673455040271529561595718940092424231253150263268513067972937042222806102175350331146290864120703025608712817763221723427454002746818270565050919821097445991953785331131470462682015972815241620750337
384  1    769
385  8  98561
386  1    773
387  2   1549
388  0    389
389  11  796673
390  3   3121
391  4   6257
392  3   3137
393  1    787
394  14  6455297
395  5  12641
396  0    397
397  4   6353
398  1    797
399  2   1597
400  0    401

Raku

First 382 in less than a second. 383 pushes the total accumulated time over 25 seconds.

-> $n { (^∞).map: -> $m { if (my $p = $n × 2 ** $m + 1).is-prime { printf "%3d %4d: %d\n",$n,$m,$p; last } } } for 1..400
Output:
  1    0: 2
  2    0: 3
  3    1: 7
  4    0: 5
  5    1: 11
  6    0: 7
  7    2: 29
  8    1: 17
  9    1: 19
 10    0: 11
 11    1: 23
 12    0: 13
 13    2: 53
 14    1: 29
 15    1: 31
 16    0: 17
 17    3: 137
 18    0: 19
 19    6: 1217
 20    1: 41
 21    1: 43
 22    0: 23
 23    1: 47
 24    2: 97
 25    2: 101
 26    1: 53
 27    2: 109
 28    0: 29
 29    1: 59
 30    0: 31
 31    8: 7937
 32    3: 257
 33    1: 67
 34    2: 137
 35    1: 71
 36    0: 37
 37    2: 149
 38    5: 1217
 39    1: 79
 40    0: 41
 41    1: 83
 42    0: 43
 43    2: 173
 44    1: 89
 45    2: 181
 46    0: 47
 47  583: 1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
 48    1: 97
 49    2: 197
 50    1: 101
 51    1: 103
 52    0: 53
 53    1: 107
 54    1: 109
 55    4: 881
 56    1: 113
 57    2: 229
 58    0: 59
 59    5: 1889
 60    0: 61
 61    4: 977
 62    7: 7937
 63    1: 127
 64    2: 257
 65    1: 131
 66    0: 67
 67    2: 269
 68    1: 137
 69    1: 139
 70    0: 71
 71    3: 569
 72    0: 73
 73    2: 293
 74    1: 149
 75    1: 151
 76    4: 1217
 77    3: 617
 78    0: 79
 79    2: 317
 80    3: 641
 81    1: 163
 82    0: 83
 83    1: 167
 84    2: 337
 85    4: 1361
 86    1: 173
 87    2: 349
 88    0: 89
 89    1: 179
 90    1: 181
 91    8: 23297
 92    7: 11777
 93    2: 373
 94  582: 1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
 95    1: 191
 96    0: 97
 97    2: 389
 98    1: 197
 99    1: 199
100    0: 101
101    3: 809
102    0: 103
103   16: 6750209
104    5: 3329
105    1: 211
106    0: 107
107    3: 857
108    0: 109
109    6: 6977
110    3: 881
111    1: 223
112    0: 113
113    1: 227
114    1: 229
115    2: 461
116    1: 233
117    3: 937
118    4: 1889
119    1: 239
120    1: 241
121    8: 30977
122    3: 977
123    6: 7873
124    6: 7937
125    1: 251
126    0: 127
127    2: 509
128    1: 257
129    3: 1033
130    0: 131
131    1: 263
132    4: 2113
133    4: 2129
134    1: 269
135    1: 271
136    0: 137
137    3: 1097
138    0: 139
139    2: 557
140    1: 281
141    1: 283
142    2: 569
143   53: 1288029493427961857
144    2: 577
145    6: 9281
146    1: 293
147    8: 37633
148    0: 149
149    3: 1193
150    0: 151
151    4: 2417
152    3: 1217
153    1: 307
154    2: 617
155    1: 311
156    0: 157
157    8: 40193
158    1: 317
159    6: 10177
160    2: 641
161    3: 1289
162    0: 163
163    2: 653
164    9: 83969
165    1: 331
166    0: 167
167    7: 21377
168    1: 337
169    2: 677
170    3: 1361
171    8: 43777
172    0: 173
173    1: 347
174    1: 349
175    2: 701
176    1: 353
177    2: 709
178    0: 179
179    1: 359
180    0: 181
181    4: 2897
182    7: 23297
183    1: 367
184    6: 11777
185    3: 1481
186    1: 373
187    6: 11969
188  581: 1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
189    1: 379
190    0: 191
191    1: 383
192    0: 193
193    2: 773
194    1: 389
195    4: 3121
196    0: 197
197   15: 6455297
198    0: 199
199    2: 797
200    1: 401
201    3: 1609
202    2: 809
203   13: 1662977
204    1: 409
205    2: 821
206   15: 6750209
207    2: 829
208    4: 3329
209    1: 419
210    0: 211
211   20: 221249537
212    3: 1697
213    2: 853
214    2: 857
215    1: 431
216    1: 433
217   66: 16011773855979890802689
218    5: 6977
219    1: 439
220    2: 881
221    1: 443
222    0: 223
223    8: 57089
224    1: 449
225    3: 1801
226    0: 227
227   11: 464897
228    0: 229
229    6: 14657
230    1: 461
231    1: 463
232    0: 233
233    1: 467
234    2: 937
235    2: 941
236    3: 1889
237    4: 3793
238    0: 239
239    1: 479
240    0: 241
241   36: 16561393893377
242    7: 30977
243    1: 487
244    2: 977
245    1: 491
246    5: 7873
247    6: 15809
248    5: 7937
249    1: 499
250    0: 251
251    1: 503
252    2: 1009
253    2: 1013
254    1: 509
255    2: 1021
256    0: 257
257  279: 249632952651006185613150855026822179503549278818199928480857894651449200648869292015617
258    2: 1033
259   38: 71193377898497
260    1: 521
261    1: 523
262    0: 263
263   29: 141197049857
264    3: 2113
265    2: 1061
266    3: 2129
267    2: 1069
268    0: 269
269    3: 2153
270    0: 271
271    4: 4337
272   11: 557057
273    1: 547
274    2: 1097
275    7: 35201
276    0: 277
277    2: 1109
278    1: 557
279    2: 1117
280    0: 281
281    1: 563
282    0: 283
283   30: 303868936193
284    1: 569
285    1: 571
286   52: 1288029493427961857
287    3: 2297
288    1: 577
289   10: 295937
290    5: 9281
291    4: 4657
292    0: 293
293    1: 587
294    7: 37633
295    2: 1181
296    1: 593
297    3: 2377
298    2: 1193
299    1: 599
300    1: 601
301    4: 4817
302    3: 2417
303    1: 607
304    2: 1217
305    3: 2441
306    0: 307
307    2: 1229
308    1: 617
309    1: 619
310    0: 311
311    9: 159233
312    0: 313
313    4: 5009
314    7: 40193
315    1: 631
316    0: 317
317    7: 40577
318    5: 10177
319    2: 1277
320    1: 641
321    1: 643
322    2: 1289
323    1: 647
324    2: 1297
325    2: 1301
326    1: 653
327    3: 2617
328    8: 83969
329    1: 659
330    0: 331
331    4: 5297
332    3: 2657
333    5: 10657
334    6: 21377
335   19: 175636481
336    0: 337
337    4: 5393
338    1: 677
339    3: 2713
340    2: 1361
341    1: 683
342    7: 43777
343    2: 1373
344    3: 2753
345    1: 691
346    0: 347
347    3: 2777
348    0: 349
349   10: 357377
350    1: 701
351   12: 1437697
352    0: 353
353   21: 740294657
354    1: 709
355    6: 22721
356    5: 11393
357    2: 1429
358    0: 359
359    1: 719
360    6: 23041
361   28: 96905199617
362    3: 2897
363    1: 727
364    6: 23297
365    5: 11681
366    0: 367
367   12: 1503233
368    5: 11777
369    1: 739
370    2: 1481
371    1: 743
372    0: 373
373    2: 1493
374    5: 11969
375    1: 751
376  580: 1487939695262196876907983166454197495251350196192890428923003345454869706240895712896623468784438158657419591298913094265537812046389415279164757669092989298186306341246574002177
377   11: 772097
378    0: 379
379   14: 6209537
380    1: 761
381    3: 3049
382    0: 383
383 6393: 11693945185971565896920916176753769281418376445302724140914106576604960252116205468905429628661873192664799900323401294531072465400997845029722990758855393414014415817179228695517839305455702961095094596926622802342799137107509767542153683280899327558274011281588755909890607960835140712630830933978801393590855371457894042968287926562847826310125559303901351824980311279986492793008248059208985097459095049075732193161126922389950080848742183055141518931962329796357335158955758486061360294773463111842316561192036096585088267052290025273980611139612478214293303564141730470933187279751846912161098280963960686648202780382930927114525552446602357404550468641236474238897222372272898562140228039886991631673186995098587756569010989657598363351856992206826342175536967926902668804937341514786382018872919876784539436965319822540039220122728568129762675989071883516915894567537630751801497223803135172643203770169327233350522822938630733126833423559124391441973547309619943019237705312515304113424366223388373606440335025932390399945086075175009569272136997988977568262327875607690344516747889133920438003737328060362069562108376086129279385800262195985974144460914705464874882401864174074796383557151951711000378565395148939760434428093058777242253682181813425273399277638142811972296863003382484684788329148214958434057306251885787781329925372401240556666727438408378656900945061970219566055969587385482421092779185798692904507774583223151161566406541599486350580593707153172641891804260963429951215526999443852964537303345106153870841180251403751871193132336680841124129779119999935597712685839886558769823834654994044516702436738265181698869580022472787153167463772595005393815295009535991557511340157179280662197799109181549751673455040271529561595718940092424231253150263268513067972937042222806102175350331146290864120703025608712817763221723427454002746818270565050919821097445991953785331131470462682015972815241620750337
384    1: 769
385    8: 98561
386    1: 773
387    2: 1549
388    0: 389
389   11: 796673
390    3: 3121
391    4: 6257
392    3: 3137
393    1: 787
394   14: 6455297
395    5: 12641
396    0: 397
397    4: 6353
398    1: 797
399    2: 1597
400    0: 401

RPL

As big ints are limited to 499 digits, it is not possible to calculate the 47th prime number.

Works with: HP version 49
« { }
  1 45 FOR n 
     0
     WHILE 2 OVER ^ n * 1 + DUP ISPRIME? NOT REPEAT 
        DROP 1 + 
     END 
     NIP +
  NEXT
» 'TASK' STO
Output:
1: { 2 3 7 5 11 7 29 17 19 11 23 13 53 29 31 17 137 19 1217 41 43 23 47 97 101 53 109 29 59 31 7937 257 67 137 71 37 149 1217 79 41 83 43 173 89 181 }

Sidef

Translation of: Perl

Takes ~2 seconds to run.

for n in (1..400) {
    var p = (^Inf -> lazy.map {|m| [m, n * 2**m + 1] }.first_by { .tail.is_prime })
    printf("%3s %4s: %s\n", n, p...)
}

(same output as the Perl version)

Wren

Library: Wren-gmp
Library: Wren-fmt
import "./gmp" for Mpz
import "./fmt" for Fmt

System.print("  N     M    Prime")
System.print("------------------")
for (n in 1..400) {
    var m = 0
    while (true) {
        var p = Mpz.from(n).mul(Mpz.one.lsh(m)).add(1)
        if (p.probPrime(15) > 0) {
            Fmt.print("$3d  $4d    $20a", n, m, p)
            break
        }
        m = m + 1
    }
}
Output:

Primes with more than 40 digits have been (mercifully) abbreviated.

  N     M    Prime
------------------
  1     0    2
  2     0    3
  3     1    7
  4     0    5
  5     1    11
  6     0    7
  7     2    29
  8     1    17
  9     1    19
 10     0    11
 11     1    23
 12     0    13
 13     2    53
 14     1    29
 15     1    31
 16     0    17
 17     3    137
 18     0    19
 19     6    1217
 20     1    41
 21     1    43
 22     0    23
 23     1    47
 24     2    97
 25     2    101
 26     1    53
 27     2    109
 28     0    29
 29     1    59
 30     0    31
 31     8    7937
 32     3    257
 33     1    67
 34     2    137
 35     1    71
 36     0    37
 37     2    149
 38     5    1217
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