A373179 a(n) is the smallest n-digit integer whose digit permutations make the maximum possible number of n-digit primes.

2, 13, 149, 1237, 13789, 123479, 1235789, 12345679, 102345679, 1123456789, 10123456789, 1011233456789, 1012334567789, 10123345677899

Offset: 1

Author

Gonzalo Martínez, May 26 2024

Keywords

• nonn
• base
• more

Comments

A065851(n) is the maximum number of n-digit primes which can be made by permuting n digits.

a(n) = k is the smallest n-digit k for which A046810(k) = A065851(n).

a(n) has its relevant digits sorted and not beginning with 0 and may or may not be one of the primes (it is for n = 1 to 7, but not at n = 8).

Examples

			For n=3, A065851(3) = 4 primes are reached by permuting the digits of a(3) = 149, namely {149, 419, 491, 941}. (4 primes are also reached from 179 and 379, but they're bigger numbers.)
		

Links

• Kevin Ryde, C Code

Crossrefs

Cf. A000040, A046810, A065851, A134596, A179239.

Programs

C

/* See links. */

Python

from sympy import nextprime
from collections import Counter
def smallest(t):
    nz = "".join(sorted(c for c in t if c != "0"))
    s = "".join(t) if "0" not in t else nz[0]+"0"*t.count("0")+nz[1:]
    return int(s)
def a(n):
    c, p = Counter(), nextprime(10**(n-1))
    while p < 10**n:
        c["".join(sorted(str(p)))] += 1
        p = nextprime(p)
    m = min(c.most_common(1), key=lambda x:smallest(x[0]))
    return smallest(m[0])  # m[1] generates A065851
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, May 28 2024

Extensions

a(9)-a(11) from Michael S. Branicky, May 27 2024

a(12)-a(14) from Kevin Ryde, Jul 16 2024