A090287 Smallest prime obtained by sandwiching n between a number with identical digits, or 0 if no such prime exists. Primes of the form k n k where all the digits of k are identical.

101, 313, 727, 131, 11411, 151, 777767777, 373, 181, 191, 9109, 0, 7127, 331333, 991499, 1151, 3163, 1171, 1181, 9199, 1201, 112111, 0, 1231, 7247, 3253, 7777777777267777777777, 1111271111, 11128111, 1291, 1301, 3313, 1321, 0, 3343, 333533, 1361, 3373, 1381

Offset: 0

Author

Amarnath Murthy, Nov 29 2003

Keywords

• base
• nonn

Comments

a(n) = 0 if n is a palindrome with even number of digits. Conjecture: No other term is zero.

The conjecture is false. a(231) = 0, a(420) = 0, a(n) = 0 if 11 divides n and n has an even number of digits. a(1414) has over 2000 digits. - Chai Wah Wu, Mar 31 2015

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..365
• Chai Wah Wu, On a conjecture regarding primality of numbers constructed from prepending and appending identical digits, arXiv:1503.08883 [math.NT], 2015.
• Index entries for primes involving decimal expansion of n

Crossrefs

Cf. A010785, A338712.

Programs

Mathematica

(* f(n) defined by José de Jesús Camacho Medina in A010785. *)
lst={};f[m_]:=IntegerDigits[(m-9*Floor[(m-1)/9])*(10^Floor[(m+8)/9]-1)/9];
g[n_]:=FromDigits[Flatten[{f[m],IntegerDigits[n],f[m]}]];
Do[m=1;While[True,If[Mod[Length[IntegerDigits[n]],2]==0&&IntegerDigits[n]==Reverse[IntegerDigits[n]],
AppendTo[lst,0];Break[],If[PrimeQ[g[n]],AppendTo[lst,g[n]];Break[]]];m++],{n,25}];
lst (* Ivan N. Ianakiev, Mar 23 2015 *)

Python

from gmpy2 import is_prime, mpz, digits
def A090287(n, limit=2000):
    sn = str(n)
    if n in (231, 420, 759) or not (len(sn) % 2 or n % 11):
        return 0
    for i in range(1, limit+1):
        for j in range(1, 10, 2):
            si = digits(j, 10)*i
            p = mpz(si+sn+si)
            if is_prime(p):
                return int(p)
    else:
        return 'search limit reached.' # Chai Wah Wu, Mar 31 2015

Extensions

a(0) from Chai Wah Wu, Mar 23 2015

a(26)-a(38) from Chai Wah Wu, Mar 24 2015