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171, 277, 367, 561, 567, 18881
COMMENTS
Digits are in ascending order beginning with 1 and after 9 comes 0.
All terms must end in 1 or 7: A057137(n) is even when n is even, and divisible by 3 iff n == 0, 2, 3, 5, 6, 8 or 9 (mod 10). - M. F. Hasler, Apr 14 2024
REFERENCES
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 61, 298.
EXAMPLE
a(1) = 12345678901234567890...01234567890...012345678901 = A057137(171) is the first prime term in A057137.
MATHEMATICA
fQ[n_] := PrimeQ@ FromDigits@ Mod[Range@n, 10]; lst = {}; Do[ If[fQ@n, AppendTo[lst, n]; Print@n], {n, 10000}]; lst
IntegerLength[Select[Table[FromDigits[PadRight[{}, n, {1, 2, 3, 4, 5, 6, 7, 8, 9, 0}]], {n, 1, 1001, 2}], PrimeQ]] (* Harvey P. Dale, Feb 07 2024 *)
PROG
(PARI) N=0; for(n=1, 600, if(ispseudoprime(N=10*N+n%10), print1(n", "))) \\ Charles R Greathouse IV, May 10 2014 (Comment: Surprisingly, this is faster than calling ispseudoprime() only when n ends in 1 or 7, even when much larger N's are considered, e.g., up to 3000. - M. F. Hasler, Apr 14 2024)
CROSSREFS
Cf. A006055, A057137, A120828, A120820, A120821, A120822, A120823, A120824, A120825, A120826, A120827.
EXTENSIONS
18881 from Arjen K. Lenstra, Feb 20 2012
Primes with n consecutive ascending digits beginning with the digit three.
+10
9
COMMENTS
Digits are in ascending order beginning with 3 and after 9 comes 0.
The sequence "Primes with n consecutive descending digits beginning with the digit three" has only one term, 1 which represents the prime 3.
MATHEMATICA
fQ[n_] := PrimeQ@ FromDigits@ Mod[2+Range@n, 10]; lst = {}; Do[ If[fQ@n, AppendTo[lst, n]; Print@n], {n, 10000}]; lst
Number of digits of the smallest prime with n consecutive ascending digits and most significant digit a five.
+10
9
COMMENTS
Digits are in ascending order beginning with 5 and after 9 comes 0.
The sequence "Primes with n consecutive digits descending beginning with the digit five" has only one term, 1 which represents the prime 5.
EXAMPLE
29 is a term since the 29-digit number 56789012345678901234567890123 is a prime.
MATHEMATICA
fQ[n_] := PrimeQ@ FromDigits@ Mod[4+Range@n, 10]; lst = {}; Do[ If[fQ@n, AppendTo[lst, n]; Print@n], {n, 9000}]; lst
Primes with n consecutive digits ascending beginning with the digit two.
+10
8
1, 2, 8, 82, 118, 158, 2122, 2242, 2388
COMMENTS
Digits are in ascending order beginning with 2 and after 9 comes 0.
The sequence "Primes with n consecutive digits descending beginning with the digit two" has only one term, 1 which represents the prime 2.
EXAMPLE
8 is a term since 23456789 is a prime.
MATHEMATICA
fQ[n_] := PrimeQ@ FromDigits@ Mod[1+Range@n, 10]; lst = {}; Do[ If[fQ@n, AppendTo[lst, n]; Print@n], {n, 10000}]; lst
Primes with n consecutive ascending digits beginning with the digit four.
+10
8
COMMENTS
Digits are in ascending order beginning with 4; after 9 comes 0.
EXAMPLE
8 is a term since 45678901 is a prime.
MATHEMATICA
fQ[n_] := PrimeQ@ FromDigits@ Mod[3+Range@n, 10]; lst = {}; Do[ If[fQ@n, AppendTo[lst, n]; Print@n], {n, 10000}]; lst
Primes with n consecutive digits ascending beginning with the digit six.
+10
8
COMMENTS
Digits are in ascending order beginning with 6 and after 9 comes 0.
The sequence "Primes with n consecutive digits descending beginning with the digit six" has no terms.
EXAMPLE
6 is a term since 678901 is a prime.
MATHEMATICA
fQ[n_] := PrimeQ@ FromDigits@ Mod[5+Range@n, 10]; lst = {}; Do[ If[fQ@n, AppendTo[lst, n]; Print@n], {n, 5000}]; lst
Number of digits of primes with n consecutive digits ascending beginning with the digit seven.
+10
8
COMMENTS
Digits are in ascending order beginning with 7 and after 9 comes 0.
The sequence "Primes with n consecutive digits descending beginning with the digit seven" has only two terms below 6001, namely 1 and 5, which represent the primes 7 and 76543, respectively.
EXAMPLE
1 is here because 7 is prime.
5 is here because 78901 is prime.
15 is here because 789012345678901 is a prime.
51 is here because 789012345678901234567890123456789012345678901234567 is prime.
MATHEMATICA
fQ[n_] := PrimeQ@ FromDigits@ Mod[6+Range@n, 10]; lst = {}; Do[ If[fQ@n, AppendTo[lst, n]; Print@n], {n, 5000}]; lst
Primes with n consecutive digits ascending beginning with the digit eight.
+10
8
COMMENTS
Digits are in ascending order beginning with 8 and after 9 comes 0.
The sequence "Primes with n consecutive digits descending beginning with the digit eight" has no terms.
EXAMPLE
2 is a term since 89 is a prime.
MATHEMATICA
fQ[n_] := PrimeQ@ FromDigits@ Mod[7+Range@n, 10]; lst = {}; Do[ If[fQ@n, AppendTo[lst, n]; Print@n], {n, 5000}]; lst
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