A053546 Smallest prime containing a leading sequence of n ascending numbers.

11, 127, 1231, 12343, 123457, 12345623, 123456719, 1234567801, 1234567891, 1234567891003, 12345678910111, 12345678910111207, 1234567891011121309, 123456789101112131449, 12345678910111213141523, 123456789101112131415161, 123456789101112131415161717

Offset

1,1

Comments

The Magma Calculator (http://magma.maths.usyd.edu.au/calc/) verified each of the 100 terms in the table as prime. - Jon E. Schoenfield, Aug 24 2009

Can this sequence be proved to be infinite, a sort of Dirichlet's Theorem in reverse? - Charles R Greathouse IV, Jul 23 2011

a(369) has 1002 digits. - Michael S. Branicky, Dec 29 2025

Links

Michael S. Branicky, Table of n, a(n) for n = 1..368 (terms 1..100 from T.D. Noe)

Formula

a(n) >= A007908(n). - Michael S. Branicky, Dec 29 2025

Example

a(6) = 12345623 is the smallest prime beginning with the string 123456.

Mathematica

sp[n_]:=Module[{m=1, np}, np=NextPrime[n*10^m]; While[Floor[np/10^m] != n, m++; np=NextPrime[n*10^m]]; np]; sp/@Table[FromDigits[ Flatten[ IntegerDigits/@ Range[x]]], {x, 20}] (* Harvey P. Dale, May 22 2016 *)

Prog

(Python)
from itertools import count
from gmpy2 import is_prime, mpz
def a(n):
    t = int("".join(str(i) for i in range(1, n+1)))
    if is_prime(t): return t
    for d in count(1):
        t *= 10
        for i in range(1, 10**d, 2):
            if is_prime(t+i):
                return t+i
print([a(n) for n in range(1, 18)]) # Michael S. Branicky, Dec 29 2025

Crossrefs

Cf. A007908.

Sequence in context: A088628 A069599 A392041 * A084975 A065543 A015598

Adjacent sequences: A053543 A053544 A053545 * A053547 A053548 A053549

Keyword

base,easy,nice,nonn

Author

Enoch Haga, Jan 16 2000

Status

approved