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Search: a270339 -id:a270339
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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
OFFSET
1,2
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
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Robert Price, Mar 20 2016
EXTENSIONS
a(38) from Robert Price, Jan 16 2020
STATUS
approved
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
OFFSET
1,2
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Robert Price, Mar 23 2016
EXTENSIONS
a(38)-a(40) from Robert Price, May 21 2018
STATUS
approved
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
OFFSET
1,3
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] &]
PROG
(PARI) is(n)=isprime((8*10^n + 49)/3) \\ Charles R Greathouse IV, Feb 16 2017
KEYWORD
nonn,more
AUTHOR
Robert Price, Mar 25 2016
EXTENSIONS
a(38)-a(40) from Robert Price, May 23 2020
STATUS
approved
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
OFFSET
1,2
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).
a(40) > 3*10^5. - Robert Price, Jul 13 2023
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
KEYWORD
nonn,base,more
AUTHOR
Robert Price, Mar 26 2016
EXTENSIONS
a(37)-a(38) from Robert Price, Mar 03 2019
a(39) from Robert Price, Jul 13 2023
STATUS
approved
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
OFFSET
1,2
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] &]
PROG
(PARI) is(n)=isprime(8*10^n - 49) \\ Charles R Greathouse IV, Feb 16 2017
KEYWORD
nonn,more
AUTHOR
Robert Price, Apr 03 2016
EXTENSIONS
a(34) from Robert Price, Aug 20 2019
STATUS
approved
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
OFFSET
1,2
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
(Magma) [n: n in [1..500] | IsPrime(7*10^n + 57)]; // Vincenzo Librandi, Jul 03 2016
KEYWORD
nonn,more
AUTHOR
Robert Price, Mar 27 2016
EXTENSIONS
a(35) from Robert Price, Jun 13 2019
STATUS
approved
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
OFFSET
1,1
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] &]
PROG
(PARI) is(n)=ispseudoprime((41*10^n + 49)/9) \\ Charles R Greathouse IV, Jun 13 2017
KEYWORD
nonn,more
AUTHOR
Robert Price, Apr 17 2016
STATUS
approved
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
OFFSET
1,1
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] &]
PROG
(PARI) is(n)=ispseudoprime((5*10^n-119)/3) \\ Charles R Greathouse IV, Jun 13 2017
KEYWORD
nonn,more
AUTHOR
Robert Price, Apr 05 2016
EXTENSIONS
a(35)-a(36) from Robert Price, Mar 29 2018
STATUS
approved
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
OFFSET
1,2
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] &]
PROG
(PARI) is(n)=ispseudoprime(16*10^n+1) \\ Charles R Greathouse IV, Jun 13 2017
KEYWORD
nonn,more
AUTHOR
Robert Price, May 12 2016
STATUS
approved
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
OFFSET
1,2
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
(Magma) [n: n in [1..400] | IsPrime(7*10^n + 43)]; // Vincenzo Librandi, Jul 03 2016
KEYWORD
nonn,more
AUTHOR
Robert Price, Jul 02 2016
STATUS
approved

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