A167416 Smallest prime concatenation of the first n primes, or 0 if no such prime exists.

2, 23, 523, 2357, 112573, 11132357, 1113257317, 111317193257, 11131719223357, 0, 111317192232935317, 11131719223293157373, 1113171922329313377541, 111317192232931337415743, 11131719223293133741474357

Offset: 1

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 03 2009

Keywords

• nonn
• base

Comments

a(10) doesn't exist, because the sum of digits of the first 10 primes (2+3+5+7+(1+1)+(1+3)+(1+7)+(1+9)+(2+3)+(2+9)) = 57 is a multiple of 3.

Examples

			The only prime concatenations of the first n primes for n = 1..3 are a(1)=2, a(2)=23, and a(3)=523.
For n=4, the only prime concatenations of 2, 3, 5, and 7 are 2357, 2753, 3257, 3527, 5237, 5273, 7253, 7523; the smallest of these is a(4) = 2357.
		

References

• Richard E. Crandall and Carl Pomerance, Prime Numbers, Springer 2005.
• Paulo Ribenboim, The New Book of Prime Number Records, Springer 1996.
• A. Weil, Number theory: an approach through history, Birkhäuser 1984.

Crossrefs

Cf. A175429, A177275, A134966, A167417.

Programs

Python

from sympy import sieve, isprime
from itertools import permutations
for n in range(1, 20):
    sieve.extend_to_no(n)
    p = list(map(str, list(sieve._list)))[:n]
    mint = 10**1000
    for i in permutations(p, len(p)):
        t = int(''.join(i))
        if  t < mint and isprime(t):
            mint = t
    if mint == 10**1000:
        print(0, end = ', ')
    else:
        print(mint, end = ', ') # Gleb Ivanov, Dec 04 2021

Extensions

Keyword:full added by R. J. Mathar, Nov 11 2009

Edited by Charles R Greathouse IV, Apr 28 2010

Several terms corrected and a(11)-a(15) from Gleb Ivanov, Dec 04 2021