A338712 -id:A338712 - cp's OEIS frontend

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

Amarnath Murthy, Nov 29 2003

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

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

Cf. A010785, A338712.

(* 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 *)

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

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

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

A338366 a(n) = smallest positive number k with all digits equal such that the concatenation k||n||k is prime, or -1 if no such k exists.

Original entry on oeis.org

1, 3, 7, 1, 11, 1, 7777, 3, 1, 1, 9, -1, 7, 33, 99, 1, 3, 1, 1, 9, 1, 11, -1, 1, 7, 3, 7777777777, 1111, 111, 1, 1, 3, 1, -1, 3, 33, 1, 3, 1, 77777777777777, 111, 3, 1111111111111111111111111111111111111111, 3, -1, 1, 3, 1, 1, 999, 7, 1, 11, 1, 7, -1, 33, 1, 3, 3, 1, 3, 1

Offset: 0

N. J. A. Sloane, Nov 08 2020

See A090287 for more information.
From Robert Price, Sep 20 2023: (Start)
For a(366), k is a string of 8441 1's.
The sequence then continues: 77, 1, 1, 3, 1, 1, 9, 7777777, 1, 11, 3, 1, 11, 9, 77, 11111, 1, 1, 33333, 3, 7, 9, 3, 1, 77, 1, 1, 9, 7777777777 until a(396) where k is a sequence of 269 1's.
The sequence then continues: 9, 777, 11, 9, 1, 7, 3, 7, 1, 11, 1, 1, 9, 9, 1111, 3, 999, 77777, 99, 7, 7, 3, 7, -1, 3, 1, 11, 77, 1, 77, 3, 1, 7, 3, 3, 1, 111111, 1, 7, 99, 7, 1111, 9, 1, 1, 11, 1, 7777777, 11, 1, 1111, 3, 1111, 7, 3, 7, 11, 3, 1, 1, 111, 3, 1, 3, 3, 1, 33, 9, 11, 33, 3, 7, 3, 3, 7, 99, 1, 1, 11, 3, 1, 9, 7, 77, 9, 1, 1, 3, 1, 7777, 33, 3, 1, 33, 3, 77, 77, 9, 1, 3, 33, 11111, 9, 9. (End)

			a(3) = 1 because 131 is prime.
a(4) = 11 because 11411 is prime, and all of 141, 242, 343, ..., 949 are composite.

Robert Price, Table of n, a(n) for n = 0..365
Index entries for primes involving decimal expansion of n

Cf. A090287.

Related sequences: A010785, A068695, A091088, A228323, A228325, A336893, A338712 (see also the Index link above).

More terms from Alois P. Heinz, Nov 08 2020

cp's OEIS Frontend

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.

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