A376027 a(n) is the least k such that the concatenations of k^i, i=0..j, are prime for j from 1 to n.

1, 3, 22557, 354081, 419163, 286623930087

Offset

1,2

Comments

For n >= 2, a(n) == 3 (mod 6) if it exists.

a(6) > 3 * 10^9 if it exists.

No further terms < 10^11, testing all odd numbers (see Python program). - Michael S. Branicky, Sep 08 2024

Links

Table of n, a(n) for n=1..6.

Example

a(3) = 22557: 22557^2 = 508818249, 22557^3 = 11477413242693, and the concatenations 122557, 122557508818249, and 12255750881824911477413242693 are all prime.

Maple

tcat:= (a, b) -> a*10^(1+ilog10(b))+b:
k:= 2: R:= 1:
for n from 2 to 5 do
 for j from k by 6 do
   c:= 1; good:= true;
   for i from 1 to n while good do
     c:= tcat(c, j^i);
     if not isprime(c) then good:= false; break fi
   od;
   if good then k:= j; R:= R, j; break fi
od od:
R;

Prog

(Python)
from itertools import count
from gmpy2 import is_prime, mpz, digits
def f(k):
    i, s = 0, "1"
    while True:
        ski = digits(k**(i+1))
        if not is_prime(mpz(s+ski)):
            break
        i, s = i+1, s+ski
    return i
def afind():
    adict = dict()
    for m in count(1, 2): # or count(3, 6) for a(2) on per comment
        v = f(m)
        if v > 0 and v not in adict:
            adict[v] = m
            print("FOUND", m, v)
afind() # Michael S. Branicky, Sep 08 2024

Crossrefs

Cf. A096469.

Sequence in context: A128121 A112406 A344681 * A171364 A307670 A115475

Adjacent sequences: A376024 A376025 A376026 * A376028 A376029 A376030

Keyword

nonn,base,more

Author

Robert Israel, Sep 06 2024

Extensions

a(6) from Michael S. Branicky, Sep 09 2024

Status

approved