A232129 -id:A232129 - cp's OEIS frontend

A232128 Maximal number of digits that can be appended to n such that each step yields a prime.

9, 7, 7, 6, 7, 6, 7, 2, 3, 10, 1, 6, 6, 4, 3, 2, 3, 5, 8, 0, 3, 5, 6, 6, 5, 5, 6, 2, 6, 2, 3, 0, 7, 5, 6, 5, 6, 3, 1, 11, 1, 4, 5, 4, 1, 7, 3, 4, 6, 4, 0, 5, 0, 6, 4, 4, 6, 2, 6, 7, 5, 0, 4, 2, 3, 3, 5, 2, 5, 4, 4, 1, 6, 2, 4, 4, 1, 7, 1, 4, 4, 10, 1, 0, 5, 1, 6, 5, 0, 1, 4

Offset: 1

M. F. Hasler, Nov 19 2013

The digits are to be appended one by one as to form a chain of L = a(n) primes [p^(1),...,p^(L)], such that p^(k-1)=floor(p^(k)/10), k=1,...,L, starting from the initial value p^(0) = n which is not required to be a prime. (See A232127 for the variant restricted to prime "starting values".)

See A232129 for the largest prime obtained when starting with n.

			a(1)=9 because "1" can be extended with at most 9 digits to the right such that each extension is prime; the least one of the possible 1+9 digit primes is 1979339333, the largest one is given in A232129.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A232125, A232127, A232129.

f:= proc(n) local V,k;
      V:= select(isprime, [seq(10*n+k, k=[1,3,7,9])]);
      if V = [] then 0 else 1 + max(map(procname,V)) fi
end proc:
map(f, [$1..100]); # Robert Israel, Oct 16 2020

a(n)=my(m,r=[0,n]);forstep(d=1,9,2,d==5&&next;isprime(n*10+d)||next;m=[1,0]+howfar(10*n+d);m[1]>r[1]&&r=m);r \\ Note: this returns the list [a(n), minimal longest prime]

from sympy import isprime, nextprime
def a(n):
    while True:
        extends, reach, maxp = -1, {n}, 0
        while len(reach) > 0:
            candidates = (int(str(e)+d) for d in "1379" for e in reach)
            reach1 = set(filter(isprime, candidates))
            extends, reach, maxp = extends+1, reach1, max({maxp}|reach1)
        return extends
print([a(n) for n in range(1, 92)]) # Michael S. Branicky, Sep 07 2021

If a(n) > 0, then there is some prime p in the range 10n+1,...,10n+9 such that a(p)=a(n)-1. If a(n)=0, then there is no prime in that range.

A231426 Least prime such that at most n digits may be appended to the right, preserving primality at each step.

Original entry on oeis.org

53, 11, 97, 17, 71, 43, 13, 2, 19, 103, 409, 1457011, 2744903797, 5232781111

Offset: 0

Michel Marcus and M. F. Hasler, Nov 19 2013

			a(7) = 2 is the least prime which starts several sequences of 1+7 primes, e.g., (2, 23, 239, 2393, ..., 23399339) and others leading at most to 29399999 = A232129(2), where a digit is appended 7 times to yield a prime after each step, while it is not possible in any of the "branches" to append one more digit to the last term, preserving primality.

Cf. A232128, A232127, A232126, A232125.

from sympy import isprime, nextprime
def a(n):
    p = 2
    while True:
        extends, reach = -1, {p}
        while len(reach) > 0:
            candidates = (int(str(e)+d) for d in "1379" for e in reach)
            reach1 = set(filter(isprime, candidates))
            extends, reach = extends + 1, reach1
        if extends == n: return p
        p = nextprime(p)
for n in range(12): print(a(n), end=", ") # Michael S. Branicky, Aug 15 2021

a(n) is the least prime p=prime(k) such that A232128(p) (= A232127(k)) = n.
If a(n) = p, then a(n) = floor(A232129(p)/10^A232128(p)).
A232125(n) <= (a(n)+1)*10^n - 1. - Michael S. Branicky, Aug 15 2021

a(12)-a(13) from Michael S. Branicky, Aug 15 2021

cp's OEIS Frontend

A232128 Maximal number of digits that can be appended to n such that each step yields a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Python

Formula