A389720 a(n) is the largest prime number obtainable by inserting a single digit anywhere in n (including at the beginning or end), or -1 if no such prime is obtainable.

7, 71, 29, 83, 47, 59, 67, 97, 89, 97, 109, 911, 127, 613, 149, 157, 167, 617, 181, 919, -1, 821, 229, 823, 241, 257, 269, 827, 283, 929, 307, 631, -1, 733, 349, 359, 367, 937, 389, 839, 409, 941, 421, 743, 449, 457, 467, 947, 487, 499, 509, 751, 523, 953, 547

Offset

0,1

Comments

We allow prepending by zero, so the fixed points here are given by A125001.

The indices of the -1 terms are a subsequence of A124665, with the first difference arising from A124665(114) = 891 and a(891) = 8971.

Links

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Example

a(1) = 71 since it is the largest of the primes 11, 13, 17, 19, 31, 41, 61, 71 obtainable by prepending or appending a single digit.
a(69) = 769 since it is the largest of the primes 269, 569, 619, 659, 691, 769 obtainable by prepending, inserting, or appending a single digit.

Maple

f:= proc(n) local i, a, b, d, x, m;
  m:= -infinity;
  for i from 0 to 1+ilog10(n) do
    b:= n mod 10^i;
    a:= (n-b)/10^i;
    for d from 9 by -1 to `if`(a=0, 1, 0) do
      x:= (10*a+d)*10^i + b;
      if x <= m then break fi;
      if isprime(x) then m:= x; break fi;
    od
  od;
  subs(-infinity=-1, m)
end proc:
f(0):= 7:
map(f, [$0..100]); # Robert Israel, Oct 12 2025

Prog

(Python)
from sympy import isprime
def a(n):
    s = str(n)
    return max((p for i in range(len(s)+1) for d in "0123456789" if isprime(p:=int(s[:i]+d+s[i:]))), default=-1)
print([a(n) for n in range(55)])

Crossrefs

Cf. A000040, A032352, A065502, A097722, A124665, A125001.

Cf. A069246, A215417, A215419, A215420, A215421.

Cf. A068679, A158594, A069832, A069833.

Cf. A389721, A389722, A389723, A389724, A389725, A389726, A389727.

Sequence in context: A090647 A200929 A111462 * A045713 A090155 A326740

Adjacent sequences: A389717 A389718 A389719 * A389721 A389722 A389723

Keyword

sign,base

Author

Michael S. Branicky, Oct 12 2025

Status

approved