20190417, 10:04  #276 
Nov 2016
2·1,249 Posts 
.........

20190417, 10:05  #277 
Nov 2016
100111000010_{2} Posts 
searched up to 20 (decimal 24) digits

20190422, 07:52  #278 
Nov 2016
2·1,249 Posts 
There are 59 (69 in decimal) bases n<=1000 (1728 in decimal) without primes of the form 2*n^k1 with k<=1000 (1728 in decimal):
{8E, 122, 175, 192, 1E0, 1E2, 20X, 213, 278, 27X, 27E, 36E, 376, 39E, 3E8, 402, 405, 412, 436, 443, 45E, 472, 479, 4X2, 4X8, 4E8, 555, 5X5, 608, 662, 688, 689, 6X5, 6X8, 70E, 738, 74E, 752, 75E, 765, 800, 835, 83E, 852, 855, 856, 864, 8XE, 918, 91X, 92E, 942, 972, 982, 9E5, X3E, X72, X95, XXE, E09, E0E, E24, E28, E52, E5E, E72, E78, EE8, EEE} Last fiddled with by sweety439 on 20190422 at 08:05 
20190422, 07:58  #279  
Nov 2016
2·1,249 Posts 
Quote:
There are.. 2 bases end with 0: 1E0, 800 no bases end with 1 14 bases end with 2: 122, 192, 1E2, 402, 412, 472, 4X2, 662, 752, 852, 942, 972, 982, X72, E52, E72 2 bases end with 3: 213, 443 2 bases end with 4: 864, E24 X bases end with 5: 175, 405, 555, 5X5, 6X5, 765, 835, 855, 9E5, X95 3 bases end with 6: 376, 436, 856 no bases end with 7 10 bases end with 8: 278, 3E8, 4X8, 4E8, 608, 688, 6X8, 738, 918, E28, E78, EE8 3 bases end with 9: 479, 689, E09 3 bases end with X: 20X, 27X, 91X 14 bases end with E: 8E, 27E, 36E, 39E, 45E, 70E, 74E, 75E, 83E, 8XE, 92E, X3E, XXE, E0E, E5E, EEE Last fiddled with by sweety439 on 20190422 at 08:13 

20190422, 08:05  #280 
Nov 2016
2·1,249 Posts 
There are 71 (85 in decimal) bases n<=1000 (1728 in decimal) without primes of the form 2*n^k+1 with k<=1000 (1728 in decimal):
{32, 85, 11E, 152, 162, 178, 195, 1EE, 215, 255, 265, 268, 27E, 28E, 2X8, 325, 32E, 368, 372, 392, 402, 442, 44E, 45E, 49E, 4X2, 528, 532, 545, 558, 569, 598, 5X2, 605, 612, 61E, 62E, 638, 642, 645, 658, 66E, 676, 6E8, 738, 742, 755, 75E, 762, 775, 7XE, 7E2, 82E, 835, 868, 892, 8X5, 8E2, 90E, 915, 932, 935, 938, 96E, 972, 98E, 998, 9E5, X02, X05, X15, X2E, X3E, X42, X58, X82, X95, XXE, XE8, E08, E3E, E42, E95, EE2, EEE} (not consider the bases end with 1, 4, 7 or X (i.e. = 1 mod 3), since for these bases n, all numbers of the form 2*n^k+1 are divisible by 3) Last fiddled with by sweety439 on 20190422 at 10:29 
20190422, 08:07  #281  
Nov 2016
2×1,249 Posts 
Quote:
There are.. no bases end with 0 no bases end with 3 1 base ends with 6: 676 1 base ends with 9: 569 

20190422, 08:10  #282  
Nov 2016
9C2_{16} Posts 
Quote:
There are 1X such bases: 11E, 1EE, 27E, 28E, 32E, 44E, 45E, 49E, 61E, 62E, 66E, 75E, 7XE, 82E, 90E, 96E, 98E, X2E, X3E, XXE, E3E, EEE Last fiddled with by sweety439 on 20190422 at 08:11 

20190422, 08:25  #283  
Nov 2016
2·1,249 Posts 
Quote:
1EE = 7 × 35 27E is prime 28E = 5 × 67 32E is prime 44E = 5 × X7 45E is prime 49E = 5 × E7 61E is prime 62E = 25 × 27 66E is prime 75E = 11 × 6E 7XE = 15 × 57 82E is prime 90E is prime 96E = 7 × 145 98E = 1E × 51 X2E = 5^{2} × 4E X3E is prime XXE is prime E3E = 7 × 175 EEE = E × 111 

20190422, 10:26  #284 
Nov 2016
2·1,249 Posts 
There are 11 (decimal 13) bases n<=1000 (decimal 1728) with both sides (2*n^k1 and 2*n^k+1) remaining:
{27E, 402, 45E, 4X2, 738, 75E, 835, 972, 9E5, X3E, X95, XXE, EEE} Last fiddled with by sweety439 on 20190422 at 10:27 
20190603, 07:48  #285 
Nov 2016
2·1,249 Posts 
.......

20190612, 03:33  #286 
Nov 2016
2·1,249 Posts 
the title should be "least k such that a063994(k)=n"
searched up to k=10^6, for n up to 400 