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

1,1

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.

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.

Links

Table of n, a(n) for n=1..15.

Example

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.

Prog

(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

Crossrefs

Cf. A175429, A177275, A134966, A167417.

Sequence in context: A360239 A054260 A053160 * A053066 A167417 A053161

Adjacent sequences: A167413 A167414 A167415 * A167417 A167418 A167419

Keyword

nonn,base

Author

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

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

Status

approved