A090287 -id:A090287 - cp's OEIS frontend

A258372 Smallest nonnegative number k not starting or ending with the digit 1 that forms a prime when it is sandwiched between n ones to the left of k and n ones to the right of k.

0, 3, 4, 8, 36, 8, 5, 72, 28, 6, 79, 212, 23, 6, 73, 24, 52, 62, 3, 28, 220, 53, 75, 58, 228, 9, 265, 89, 214, 86, 215, 4, 7, 39, 295, 40, 87, 216, 97, 6, 264, 53, 287, 223, 4, 239, 259, 25, 57, 364, 49, 38, 93, 86, 27, 30, 80, 24, 6, 356, 50, 645, 395, 206

Offset: 1

Felix Fröhlich, May 28 2015

n = 1 is the only case where a(n) = 0, since for any n > 1, A138148(n) is divisible by A002275(n).
No n exists such that a(n) = 2, since any number of the form A100706(n)+A011557(n) is of the form A000533(n)*A002275(n+1) (see comment by Robert Israel in A107123).
a(n) = 3 iff n is in A107123.
a(n) = 4 iff n is in A107124.
If k has an even number of digits and is a multiple of 11, then k is not a term. If k = (10^r+1)(10^m-1)/9 for some m > 0, r >= 0, then k is not a term. If A272232(k) = 0, then k is not a term. - Chai Wah Wu, Nov 08 2019

			a(1) = 0, because 101 is prime.
a(5) = 36, because the smallest x >= 0 such that 11111_x_11111 (where '_' denotes concatenation) is prime is 36. The decimal expansion of that prime is 111113611111.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A088281, A090287, A272232.

Table[k = 0; s = Table[1, {n}]; While[Or[!PrimeQ[FromDigits[s ~Join~ IntegerDigits[k] ~Join~ s]], Or[First@ IntegerDigits@ k == 1, Last@ IntegerDigits@ k == 1]], k++]; k, {n, 64}] (* Michael De Vlieger, May 28 2015 *)

a000042(n) = (10^n-1)/9
a(n) = my(k=0); while(k==10 || k%10==1 || k\(10^(#Str(k)-1))==1 || !ispseudoprime(eval(Str(a000042(n), k, a000042(n)))), k++); k

A256481 Smallest prime obtained by appending a number with identical digits to n or 0 if no such prime exists.

Original entry on oeis.org

2, 11, 23, 31, 41, 53, 61, 71, 83, 97, 101, 113, 127, 131, 149, 151, 163, 173, 181, 191, 2011, 211, 223, 233, 241, 251, 263, 271, 281, 293, 307, 311, 3299, 331, 347, 353, 367, 373, 383, 397, 401, 419, 421, 431, 443, 457, 461, 479, 487, 491, 503, 511111, 521, 5333

Offset: 0

Chai Wah Wu, Mar 31 2015

For n <= 15392, a(n) = 0 if and only if n = 6930. Conjecture: if a(n) = 0, then n is divisible by 3. Conjecture verified for n <= 10^6. a(n) = 0 for n = 6930, 50358, 56574, 72975.

Chai Wah Wu, Table of n, a(n) for n = 0..6068
Chai Wah Wu, On a conjecture regarding primality of numbers constructed from prepending and appending identical digits, arXiv:1503.08883 [math.NT], 2015.

Cf. A090287, A256480, A030665.

from gmpy2 import mpz, digits, is_prime
def A256481(n,limit=2000):
    if n in (6930,50358,56574,72975):
        return 0
    if n == 0:
        return 2
    sn = str(n)
    for i in range(1,limit+1):
        for j in range(1,10,2):
            si = digits(j,10)*i
            p = mpz(sn+si)
            if is_prime(p):
                return int(p)
    else:
        return 'search limit reached.'

A338712 Numbers with all digits equal and from the set {1, 3, 7, 9}.

Original entry on oeis.org

1, 3, 7, 9, 11, 33, 77, 99, 111, 333, 777, 999, 1111, 3333, 7777, 9999, 11111, 33333, 77777, 99999, 111111, 333333, 777777, 999999, 1111111, 3333333, 7777777, 9999999, 11111111, 33333333, 77777777, 99999999, 111111111, 333333333, 777777777, 999999999, 1111111111, 3333333333, 7777777777, 9999999999

Offset: 1

N. J. A. Sloane, Nov 08 2020, following a suggestion from Hugo Pfoertner

Candidates for prefixes and suffixes in A090287.

Robert Price, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,11,0,0,0,-10).

Cf. A065444, A090287, A338366.

a:= n-> [1, 3, 7, 9][1+irem(n-1, 4)]*(10^iquo(n+3, 4)-1)/9:
seq(a(n), n=1..50);  # Alois P. Heinz, Nov 09 2020

A338712={}; Do[AppendTo[A338712, FromDigits[ConstantArray[#,i]] & /@{ 1,3,7,9}], {i,10}]; A338712//Flatten (* Robert Price, Sep 21 2023 *)

From Colin Barker, Nov 09 2020: (Start)
G.f.: x*(1 + 3*x + 7*x^2 + 9*x^3) / ((1 - x)*(1 + x)*(1 + x^2)*(1 - 10*x^4)).
a(n) = 11*a(n-4) - 10*a(n-8) for n>8. (End)
Sum_{n>=1} 1/a(n) = (100/63) * A065444. - Amiram Eldar, Aug 31 2025

A256048 Smallest palindromic prime by adding at least one digit to both the left and right of n.

Original entry on oeis.org

101, 313, 727, 131, 11411, 151, 10601, 373, 181, 191, 30103, 1114111, 1120211, 11311, 11411, 31513, 1160611, 1117111, 18181, 71917, 30203, 1120211, 72227, 32323, 12421, 1250521, 36263, 12721, 12821, 39293, 10301, 11311, 32323, 13331, 14341, 33533, 16361, 77377

Offset: 0

Felix Fröhlich, Mar 10 2015

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A090287

from _future_ import division
from sympy import isprime
def palgenrange2(m,l): # generator of odd-length palindromes of length at least m and at most 2*l
    if m == 1:
        yield 0
    for x in range(m//2+1,l+1):
        n = 10**(x-1)
        for y in range(n,n*10):
            s = str(y)
            yield int(s+s[-2::-1])
def A256048(n):
    sn = str(n)
    for p in palgenrange2(len(sn)+2,len(sn)+20):
        if sn in str(p)[1:-1] and isprime(p):
            break
    else:
        return 'search limit reached'
    return p # Chai Wah Wu, Mar 22 2015

a(10), a(12), a(16), a(18) corrected by Chai Wah Wu, Mar 22 2015

A256480 Smallest prime obtained by appending n to a nonzero number with identical digits or 0 if no such prime exists.

Original entry on oeis.org

0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 211, 0, 113, 0, 0, 0, 317, 0, 419, 0, 421, 0, 223, 0, 0, 0, 127, 0, 229, 0, 131, 0, 233, 0, 0, 0, 137, 0, 139, 0, 241, 0, 443, 0, 0, 0, 347, 0, 149, 0, 151, 0, 353, 0, 0, 0, 157, 0, 359, 0, 461, 0, 163, 0, 0, 0, 167, 0, 269

Offset: 0

Chai Wah Wu, Mar 31 2015

a(n) = 0 if n is even or a multiple of 5. Conjecture: all other terms are nonzero. Conjecture verified for n <= 10^7.

"Appending" means "on the right".

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Chai Wah Wu, On a conjecture regarding primality of numbers constructed from prepending and appending identical digits, arXiv:1503.08883 [math.NT], 2015.

Cf. A090287, A256481.

from gmpy2 import digits, mpz, is_prime
def A256480(n,limit=2000):
    sn = str(n)
    if not (n % 2 and n % 5):
        return 0
    for i in range(1,limit+1):
        for j in range(1,10):
            si = digits(j,10)*i
            p = mpz(si+sn)
            if is_prime(p):
                return int(p)
    else:
        return 'search limit reached.'

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

A252942 Smallest prime of the form "Concatenate(m,n,m)".

Original entry on oeis.org

101, 313, 727, 131, 11411, 151, 13613, 373, 181, 191, 9109, 131113, 7127, 171317, 131413, 1151, 3163, 1171, 1181, 9199, 1201, 112111, 172217, 1231, 7247, 3253, 372637, 172717, 232823, 1291, 1301, 3313, 1321, 233323, 3343, 273527, 1361, 3373, 1381, 173917, 174017

Offset: 0

Ivan N. Ianakiev, Mar 23 2015

			111 is divisible by 3, and 212 is divisible by 2, but 313 is prime; therefore, a(1) = 313.

Danny Rorabaugh, Table of n, a(n) for n = 0..10000

Cf. A090287, A256048.

Cf. A010051.

a252942 n = head [y | m <- [1..],
   let y = read (show m ++ show n ++ show m) :: Integer, a010051' y == 1]
-- Reinhard Zumkeller, Apr 08 2015

f:= proc(n) local dn, x, dx,p;
  dn:= 10^(1+ilog10(n));
  for x from 1 by 2 do if igcd(x,n) = 1 then
     dx:= 10^(1+ilog10(x));
     p:= x*(1+dx*dn)+n*dx;
     if isprime(p) then return(p) fi
  fi od
end proc:
101, seq(f(n), n=1..100); # Robert Israel, Apr 07 2015
# second Maple program:
a:= proc(n) local m, p; for m do
      p:= parse(cat(m, n, m));
      if isprime(p) then break fi od; p
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 16 2020

mnmPrimes = {}; f[m_, n_] := FromDigits[Flatten[{IntegerDigits[m], IntegerDigits[n], IntegerDigits[m]}]]; Do[m = 1; While[True, If[PrimeQ[f[m, n]], AppendTo[mnmPrimes, f[m, n]]; Break[]]; m+=2], {n, 0, 40}]; mnmPrimes

a(n) = {m=1; while (! isprime(p=eval(concat(Str(m), concat(Str(n), Str(m))))), m+=2); p;} \\ Michel Marcus, Mar 23 2015

def A252942(n):
    m = 1
    sn = str(n)
    while True:
        sm = str(m)
        a = int(sm + sn + sm)
        if is_prime(a):
            return a
        m += 2
A252942(40) # Danny Rorabaugh, Mar 31 2015

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A258372 Smallest nonnegative number k not starting or ending with the digit 1 that forms a prime when it is sandwiched between n ones to the left of k and n ones to the right of k.

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A256481 Smallest prime obtained by appending a number with identical digits to n or 0 if no such prime exists.

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A338712 Numbers with all digits equal and from the set {1, 3, 7, 9}.

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A256048 Smallest palindromic prime by adding at least one digit to both the left and right of n.

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A256480 Smallest prime obtained by appending n to a nonzero number with identical digits or 0 if no such prime exists.

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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.

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A252942 Smallest prime of the form "Concatenate(m,n,m)".

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