A071234 Smallest prime beginning and ending in 2n+1 or 0 if no such prime exists.

11, 3, 5, 7, 919, 11, 13, 0, 17, 19, 21121, 23, 0, 27127, 29, 31, 33533, 0, 37, 39139, 41, 43, 0, 47, 49549, 51151, 53, 0, 57457, 59, 61, 63463, 0, 67, 690269, 71, 73, 0, 77377, 79, 81181, 83, 0, 87187, 89, 91291, 93493, 0, 97

Offset

0,1

Comments

If the two copies of the 2n+1 are not allowed to share digits, a(1) is not 3, a(2) is not 5 and a(3) is not 7 and a(5) is not 11, which results in A070278.

Conjecture: a(n) = 0 if and only if n > 2 and n == 2 mod 5 (which makes 2*n+1 == 0 mod 5). - Robert Israel, May 26 2015

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Example

a(6060) = 1212121, because it is the smallest prime beginning and ending in 2*6060+1 = 12121.
a(7+5*k) = 0, because 2*(7+5*k)+1 = 15+10*k == 0 (mod 5) is no prime.

Maple

A071234 := proc(n)
        local dgsn , p, wrks, doff;
        dgsn := convert(2*n+1, base, 10) ;
        for i from 1 to 1000000 do
                p := ithprime(i) ;
                wrks := true;
                dgsp := convert(p, base, 10) ;
                doff := nops(dgsp)-nops(dgsn) ;
                if doff >= 0 then
                        for d from 1 to nops(dgsn) do
                                if op(d, dgsp) <> op(d, dgsn) then
                                        wrks := false;
                                        break;
                                end if;
                                if op(d+doff, dgsp) <> op(d, dgsn) then
                                        wrks := false;
                                        break;
                                end if;
                        end do:
                        if wrks then
                                return p
                        end if;
                end if;
        end do:
        return 0 ;
end proc: # R. J. Mathar, Feb 03 2011
# Alternative:
f:= proc(n)
    local u, d, r, x, y;
    u:= 2*n+1;
    if isprime(u) then return(u) fi;
    if u mod 5 = 0 then return(0) fi;
    d:= ilog10(u);
    for r from 0 do
      for x from 0 to 10^(r+1)-1 do
       y:= u + 10^(d+1)*x + 10^(r+d+2)*u;
      if isprime(y) then return(y) fi
    od od
end proc:
11, seq(f(n), n=1..100); # Robert Israel, May 26 2015

Prog

(Sage)
def A071234(n):
    s = str(2*n+1)
    if s.endswith('5') and not s == '5': return 0
    for p in Primes():
        ps = str(p)
        if ps.startswith(s) and ps.endswith(s): return p
[A071234(n) for n in range(20)]
# D. S. McNeil, Feb 03 2011

Crossrefs

Cf. A070278.

Sequence in context: A070720 A010189 A338716 * A082626 A344542 A334132

Adjacent sequences: A071231 A071232 A071233 * A071235 A071236 A071237

Keyword

nonn,base

Author

Amarnath Murthy, May 18 2002

Status

approved