A101115 Beginning with the n-th prime, the number of successive times a new prime can be formed by prepending the smallest nonzero digit.

0, 5, 0, 9, 5, 4, 8, 4, 5, 9, 4, 6, 2, 7, 6, 8, 9, 7, 6, 3, 14, 5, 5, 2, 4, 10, 1, 5, 7, 3, 4, 3, 5, 5, 0, 6, 5, 8, 5, 13, 4, 5, 4, 5, 3, 8, 4, 4, 5, 8, 3, 6, 1, 4, 4, 2, 5, 2, 2, 3, 4, 9, 8, 7, 4, 7, 3, 3, 5, 5, 7, 8, 4, 3, 3, 2, 1, 7, 0, 4, 3, 5, 3, 7, 9, 6, 6, 5, 6, 8

Offset: 1

Author

Chuck Seggelin (seqfan(AT)plastereddragon.com), Dec 02 2004

Keywords

• base
• nonn

Comments

It is possible the procedure described would generate some left-truncatable primes (A024785). Although zero digits cannot be added, it is possible the starting prime may contain zeros. Therefore the possible number of digit additions is not limited by the length of the largest known left-truncatable prime. Further, because the smallest digit that satisfies the requirement is used each time, it is possible that choosing a larger digit would allow more single digits to be added. Therefore although some of the set of left-truncatable primes may be generated by this practice, not all of them will.

In principle it is possible that some a(n) is undefined because the process could go on indefinitely, but this is very unlikely. The largest a(n) for n <= 300000 is a(49120) = 18. - Robert Israel, Jun 29 2015

Examples

			a(2) is 5 because the second prime is 3, to which single nonzero digits can be prepended 5 times yielding a new prime each time (giving preference to the smallest digit that satisfies the requirement): 13, 113, 2113, 12113, 612113 (see A053583). There is no nonzero digit which can be prepended to 612113 to yield a new prime.
a(21) = 14 because the 21st prime (73) can be prepended with single nonzero digits 14 times yielding a new prime each time: 73, 173, 6173, 66173, ..., 4818372912366173.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• I. O. Angell, and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
• Index entries for sequences related to truncatable primes

Crossrefs

Cf. A053583, A024785, A000040, A101116, A101117, A101118.

Programs

Maple

f:= proc(n) local p, nd, d, count, x, success;
      p:= ithprime(n); nd:= ilog10(p);
      for count from 0 do
        nd:= nd+1;
        success:= false;
        for d from 1 to 9 do
          x:= 10^nd * d + p;
          if isprime(x) then
             success:= true;
             break
          fi
        od;
        if not success then return(count) fi;
        p:= x;
      od
end proc:
map(f, [$1..100]); # Robert Israel, Jun 29 2015

Python

from sympy import isprime, prime
def a(n):
    pn = prime(n)
    s, c, found = str(pn), 0, True
    while found:
        found = False
        for d in "123456789":
            if isprime(int(d+s)):
                s, c, found = d+s, c+1, True
                break
    return c
print([a(n) for n in range(1, 91)]) # Michael S. Branicky, Jun 24 2022