Displaying 1-10 of 502 results found.
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Numbers k such that (68*10^k + 7)/3 is prime.
+10
500
1, 2, 3, 4, 7, 10, 24, 25, 29, 34, 35, 37, 46, 49, 88, 103, 290, 381, 484, 696, 751, 886, 999, 1750, 5062, 6214, 9740, 12558, 16551, 24674, 28600, 37427, 48032, 61991, 70148, 72516, 99441, 179656
COMMENTS
Numbers k such that the digits 22 followed by k-1 occurrences of the digit 6 followed by the digit 9 is prime (see Example section).
a(39) > 3*10^5.
EXAMPLE
3 is in this sequence because (68*10^3+7)/3 = 22669 is prime.
Initial terms and associated primes:
a(1) = 1, 229;
a(2) = 2, 2269;
a(3) = 3, 22669;
a(4) = 4, 226669;
a(5) = 7, 226666669, etc.
MATHEMATICA
Select[Range[0, 100000], PrimeQ[(68*10^# + 7)/3] &]
PROG
(PARI) lista(nn) = for(n=1, nn, if(ispseudoprime((68*10^n + 7)/3), print1(n, ", "))); \\ Altug Alkan, Mar 20 2016
Numbers k such that (7*10^k + 71)/3 is prime.
+10
498
1, 2, 3, 4, 5, 7, 23, 29, 37, 39, 40, 89, 115, 189, 227, 253, 301, 449, 533, 607, 969, 1036, 1207, 1407, 1701, 3493, 7147, 11342, 21638, 22327, 25575, 25648, 34079, 39974, 47719, 49913, 74729, 100737, 103531, 168067
COMMENTS
For k > 1, numbers k such that the digit 2 followed by k-2 occurrences of the digit 3 followed by the digits 57 is prime (see Example section).
a(41) > 2*10^5.
EXAMPLE
3 is in this sequence because (7*10^3 + 71)/3 = 2357 is prime.
Initial terms and associated primes:
a(1) = 1, 47;
a(2) = 2, 257;
a(3) = 3, 2357;
a(4) = 4, 23357;
a(5) = 5, 233357, etc.
MATHEMATICA
Select[Range[0, 100000], PrimeQ[(7*10^# + 71)/3] &]
PROG
(PARI) lista(nn) = {for(n=1, nn, if(ispseudoprime((7*10^n + 71)/3), print1(n, ", "))); } \\ Altug Alkan, Mar 23 2016
Numbers k such that (8*10^k + 49)/3 is prime.
+10
497
0, 1, 2, 3, 4, 5, 6, 10, 24, 33, 34, 35, 45, 52, 56, 62, 65, 103, 166, 424, 886, 1418, 1825, 4895, 5715, 7011, 7810, 9097, 12773, 14746, 20085, 25359, 27967, 46629, 48507, 68722, 74944, 102541, 118960, 157368
COMMENTS
For k > 2, numbers k such that the digit 2 followed by k-3 occurrences of the digit 6 followed by the digits 83 is prime (see Example section).
a(41) > 3*10^5.
EXAMPLE
3 is in this sequence because (8*10^3 + 49)/3 = 2683 is prime.
Initial terms and associated primes:
a(1) = 0, 19;
a(2) = 1, 43;
a(3) = 2, 283;
a(4) = 3, 2683;
a(5) = 4, 26683;
a(6) = 5, 266683, etc.
MATHEMATICA
Select[Range[0, 100000], PrimeQ[(8*10^# + 49)/3] &]
Numbers k such that (16*10^k - 31)/3 is prime.
+10
496
1, 2, 3, 4, 15, 20, 24, 32, 38, 40, 63, 93, 104, 194, 208, 514, 535, 600, 928, 1300, 1485, 1780, 2058, 3060, 3356, 3721, 6662, 11552, 15482, 23000, 27375, 34748, 57219, 61251, 85221, 99656, 103214, 103244, 276537
COMMENTS
For k > 1, numbers k such that the digit 5 followed by k-2 occurrences of the digit 3 followed by the digits 23 is prime (see Example section).
EXAMPLE
3 is in this sequence because (16*10^3 - 31)/3 = 5323 is prime.
Initial terms and associated primes:
a(1) = 1, 43;
a(2) = 2, 523;
a(3) = 3, 5323;
a(4) = 4, 53323;
a(5) = 15, 5333333333333323;
a(6) = 20, 533333333333333333323, etc.
MATHEMATICA
Select[Range[0, 100000], PrimeQ[(16*10^# - 31)/3] &]
PROG
(PARI) isok(n) = ispseudoprime((16*10^n - 31)/3); \\ Michel Marcus, Mar 26 2016
Numbers k such that 8*10^k - 49 is prime.
+10
490
1, 2, 3, 8, 24, 49, 57, 74, 104, 131, 144, 162, 182, 246, 302, 352, 557, 581, 589, 704, 939, 1181, 1937, 2157, 4463, 6013, 7266, 8504, 8691, 16129, 20108, 40677, 74234, 112018
COMMENTS
For k > 1, numbers k such that the digit 7 followed by k-2 occurrences of the digit 9 followed by the digits 51 is prime (see Example section).
a(35) > 2*10^5.
EXAMPLE
3 is in this sequence because 8*10^3 - 49 = 7951 is prime.
Initial terms and associated primes:
a(1) = 1, 31;
a(2) = 2, 751;
a(3) = 3, 7951;
a(4) = 8, 799999951;
a(6) = 24, 7999999999999999999999951, etc.
MATHEMATICA
Select[Range[0, 100000], PrimeQ[8*10^# - 49] &]
Numbers k such that 7*10^k + 57 is prime.
+10
3
1, 2, 3, 5, 6, 7, 12, 14, 19, 21, 27, 33, 60, 61, 91, 102, 535, 549, 614, 695, 709, 1014, 2448, 2519, 3464, 3511, 6348, 6841, 11009, 11177, 20754, 26610, 30651, 39246, 122294
COMMENTS
For k > 1, numbers k such that the digit 7 followed by k-2 occurrences of the digit 0 followed by the digits 57 is prime (see Example section).
a(35) > 2*10^5.
EXAMPLE
3 is in this sequence because 7*10^3+57 = 7057 is prime.
Initial terms and associated primes:
a(1) = 1, 127;
a(2) = 2, 757;
a(3) = 3, 7057;
a(4) = 5, 700057;
a(5) = 6, 7000057;
a(6) = 7, 70000057, etc.
MATHEMATICA
Select[Range[0, 100000], PrimeQ[7*10^# + 57] &]
PROG
(PARI) lista(nn) = {for(n=1, nn, if(ispseudoprime(7*10^n + 57), print1(n, ", "))); } \\ Altug Alkan, Mar 27 2016
Numbers k such that (41*10^k + 49)/9 is prime.
+10
1
2, 3, 6, 20, 26, 38, 51, 119, 155, 218, 446, 486, 1211, 1319, 1338, 1365, 1575, 5106, 7019, 9503, 9695, 14304, 15417, 17765, 24222, 25500, 26306, 35238, 93207
COMMENTS
For terms k > 1, numbers that begin with the digit 4 followed by k-2 occurrences of the digit 5 followed by the digits 61 are prime (see Example section).
a(30) > 2*10^5.
EXAMPLE
3 is in this sequence because (41*10^3 + 49)/9 = 4561 is prime.
Initial terms and associated primes:
a(1) = 2, 461;
a(2) = 3, 4561;
a(3) = 6, 4555561;
a(4) = 20, 455555555555555555561;
a(5) = 26, 455555555555555555555555561, etc.
MATHEMATICA
Select[Range[0, 100000], PrimeQ[(41*10^# + 49)/9] &]
Numbers k such that (5 * 10^k - 119)/3 is prime.
+10
1
2, 3, 5, 6, 8, 11, 26, 33, 35, 41, 69, 73, 204, 230, 295, 381, 392, 537, 776, 1187, 2187, 2426, 4182, 4589, 5841, 6107, 11513, 13431, 28901, 56256, 65203, 66613, 82085, 91707, 126871, 140281
COMMENTS
For k > 1, numbers k such that the digit 1 followed by k - 2 occurrences of the digit 6 followed by the digits 27 is prime (see Example section).
a(37) > 2*10^5.
EXAMPLE
3 is in this sequence because (5*10^3 - 119)/3 = 1627 is prime.
Initial terms and associated primes:
a(1) = 2, 127;
a(2) = 3, 1627;
a(3) = 5, 166627;
a(4) = 6, 1666627;
a(5) = 8, 166666627, etc.
MATHEMATICA
Select[Range[10^5], PrimeQ[(5 * 10^# - 119)/3] &]
Numbers k such that 16*10^k + 1 is prime.
+10
1
0, 2, 3, 4, 18, 21, 36, 58, 68, 78, 84, 94, 150, 178, 190, 591, 686, 812, 840, 2308, 2530, 2884, 4311, 6134, 7695, 8004, 8109, 9777, 15570, 17505
COMMENTS
For k > 1, numbers k such that the digits 16 followed by k-1 occurrences of the digit 0 followed by the digit 1 is prime (see Example section).
a(31) > 10^5.
EXAMPLE
4 is in this sequence because 16*10^4+1 = 160001 is prime.
Initial terms and associated primes:
a(1) = 0, 17;
a(2) = 2, 1601;
a(3) = 3, 16001;
a(4) = 4, 160001;
a(5) = 18, 16000000000000000001. etc.
MATHEMATICA
Select[Range[0, 100000], PrimeQ[16*10^#+1] &]
Numbers k such that 7*10^k + 43 is prime.
+10
1
1, 2, 3, 7, 26, 27, 36, 44, 50, 57, 59, 73, 124, 152, 154, 250, 271, 301, 376, 451, 1177, 2299, 3740, 13159, 14780, 17435, 30098, 32521
COMMENTS
For k > 1, numbers k such that the digit 7 followed by k-2 occurrences of the digit 0 followed by the digits 43 is prime (see Example section).
a(29) > 10^5.
EXAMPLE
3 is in this sequence because 7*10^3 + 43 = 7043 is prime.
Initial terms and associated primes:
a(1) = 1, 113;
a(2) = 2, 743;
a(3) = 3, 7043;
a(4) = 7, 70000043;
a(5) = 26, 700000000000000000000000043, etc.
MATHEMATICA
Select[Range[0, 100000], PrimeQ[7*10^# + 43] &]
PROG
(PARI) lista(nn) = for(n=1, nn, if(ispseudoprime(7*10^n + 43), print1(n, ", "))); \\ Altug Alkan, Jul 02 2016
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