A389927 The total length of the sequence when starting from n and creating the smallest unused prime number by either removing or adding a single digit anywhere in the value of the previous number. If the sequence does not terminate, a(n) = -1.

17608, 205696, 205694, 17612, 205696, 17614, 200955, 205696, 17621, 17608, 17607, 205697, 17604, 17622, 205697, 17606, 175096, 17609, 17620, 205697, 205697, 205697, 205710, 205697, 205697, 205697, 205697, 205697, 205697, 205695, 162055, 205697, 205695, 205695

Offset

1,1

Comments

See A389824 and A389825 for the sequence terms when starting at 1 and 2 respectively.

It is conjectured that the sequence of primes terminates for all starting values.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Formula

From Michael S. Branicky, Oct 22 2025: (Start)

a(n) = 1 + a(A389929(n)) for n composite, in particular

a(n) = 1 when A389929(n) = 0. (End)

Example

a(1) = 17608. See A389824.
a(2) = 205696. See A389825.

Prog

(Python)
from gmpy2 import is_prime
from functools import cache
def f(n, avoid):
    s = str(n)
    D = set(p for i in range(len(s)) if len(t:=s[:i]+s[i+1:]) and is_prime(p:=int(t)))
    if D and (Dcands:=D-avoid):
        return min(Dcands)
    A = set(p for i in range(len(s)+1) for d in "0123456789" if is_prime(p:=int(s[:i]+d+s[i:])))
    return min(Acands) if A and (Acands:=A-avoid) else 0
@cache
def a(n):
    if n > 1 and not is_prime(n):
        fn = f(n, set())
        return 1 if (fn:=f(n, set())) == 0 else 1 + a(fn)
    an, aset = n, {n}
    while an != 0:
        aset.add(an)
        an = f(an, aset)
    return len(aset)
# Michael S. Branicky, Oct 22 2025

Crossrefs

Cf. A389824, A389825, A389925, A389926, A387852, A389929.

Cf. A000040, A080603, A080608, A068166.

Sequence in context: A253403 A213107 A167190 * A203353 A273035 A036377

Adjacent sequences: A389924 A389925 A389926 * A389928 A389929 A389930

Keyword

nonn,base

Author

Scott R. Shannon, Oct 19 2025

Status

approved