A210511 -id:A210511 - cp's OEIS frontend

A210712 Primes formed by concatenating n, n, n, and 1 for n = 1, 2, 3,....

2221, 3331, 4441, 6661, 1212121, 1616161, 2323231, 2424241, 2828281, 3131311, 3232321, 3333331, 3636361, 3838381, 4040401, 4242421, 4545451, 5454541, 5858581, 6262621, 6868681, 6969691, 7171711, 7272721, 8282821, 9090901, 9999991, 1161161161

Offset: 1

Jonathan Vos Post, Jan 29 2013

This is to three (duplicated strings concatenated) as A210511 is to two (duplicated strings concatenated).

			a(1) = 2221 because "2" concatenated with "2" concatenated with "2" concatenated with "1" = 2221, which is prime.
9393931 is not in the sequence, even though formed by concatenating n, n, n, and 1 for n = 93, because 9393931 = 211^3 is composite.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A210511.

[nnn1: n in [1..120] | IsPrime(nnn1) where nnn1 is Seqint([1] cat Intseq(n) cat Intseq(n) cat Intseq(n))]; // Bruno Berselli, Jan 30 2013

Select[Table[FromDigits[Flatten[{IntegerDigits[n], IntegerDigits[n], IntegerDigits[n], IntegerDigits[1],{}}]], {n, 1000}], PrimeQ] (* Vincenzo Librandi, Mar 13 2013 *)
Select[FromDigits/@Table[Flatten[IntegerDigits/@PadLeft[{1},4,n]],{n,120}],PrimeQ] (* Harvey P. Dale, May 12 2015 *)

a(n) = { my(p=Str(prime(n))); eval(concat(concat(concat(p,p),p),1)); } /* Joerg Arndt, Mar 14 2013 */

A210512 Primes formed by concatenating k, k and 3 for k >= 1.

Original entry on oeis.org

113, 223, 443, 773, 883, 10103, 11113, 14143, 25253, 26263, 28283, 32323, 35353, 41413, 50503, 61613, 68683, 71713, 77773, 80803, 83833, 85853, 88883, 97973, 1001003, 1011013, 1101103, 1131133, 1161163, 1181183, 1221223, 1241243, 1281283, 1331333, 1361363, 1391393

Offset: 1

Abhiram R Devesh, Jan 26 2013

This sequence is similar to A030458, A052089 and A210511.
k must not be a multiple of 3, otherwise the concatenation of k, k and 3 will also be a multiple of 3 and therefore not prime. This is a necessary but not sufficient condition.
Some of the terms can be found with this simple process: 5 - 3 = 2 = 1 + 1 giving 113; 7 - 3 = 4 = 2 + 2 giving 223; 11 - 3 = 8 = 4 + 4 giving 443; 17 - 3 = 14 = 7 + 7 giving 773; 19 - 3 = 16 = 8 + 8 giving 883. - J. M. Bergot, Jul 25 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A030458, A052089, A210511.

[nn3: n in [1..140] | IsPrime(nn3) where nn3 is Seqint([3] cat Intseq(n) cat Intseq(n))]; // Bruno Berselli, Jan 30 2013

Select[Table[FromDigits[Flatten[{IntegerDigits[n], IntegerDigits[n], {3}}]], {n, 100}], PrimeQ] (* Alonso del Arte, Jan 27 2013 *)

import numpy as np
from functools import reduce
def factors(n):
    return reduce(list._add_, ([i, n//i] for i in range(1, int(n**0.5) +1) if n % i == 0))
for i in range(1, 1000):
    p1=int(str(i)+str(i)+"3")
    if len(factors(p1))<3:
        print(p1, end=',')

from sympy import isprime
def xf(n): return int(str(n)*2+'3')
def ok(n): return isprime(xf(n))
print(list(map(xf, filter(ok, range(1, 140))))) # Michael S. Branicky, May 21 2021

A210720 Primes formed by concatenating n, n, n, n, and 1 for n = 1, 2, 3,....

Original entry on oeis.org

33331, 99991, 242424241, 404040401, 454545451, 464646461, 494949491, 525252521, 575757571, 737373731, 787878781, 949494941, 1021021021021, 1081081081081, 1091091091091, 1211211211211, 1291291291291, 1481481481481, 1511511511511

Offset: 1

Jonathan Vos Post, Jan 29 2013

This is to four (duplicated strings concatenated) as A210712 is to three (duplicated strings concatenated), and as A210511 is to two (duplicated strings concatenated).

			a(1) = 33331 because Concat(3,3,3,1) = 3331 which is in A000040.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A210511, A210712.

[nnnn1: n in [1..200] | IsPrime(nnnn1) where nnnn1 is Seqint([1] cat Intseq(n) cat Intseq(n) cat Intseq(n) cat Intseq(n))]; // Vincenzo Librandi, Mar 15 2013

A210720 := proc(n)
        local p;
        [n,n,n,n,1] ;
        p := digcatL(%) ;
        if isprime(p) then
                printf("%d,",p) ;
        end if;
end proc:
for n from 1 to 400 do
        A210720(n) ;
end do: # R. J. Mathar, Feb 10 2013

Select[Table[FromDigits[Flatten[{IntegerDigits[n], IntegerDigits[n], IntegerDigits[n], IntegerDigits[n], IntegerDigits[1], {}}]], {n, 200}], PrimeQ] (* Vincenzo Librandi, Mar 15 2013 *)
Select[Table[FromDigits[Join[Flatten[IntegerDigits/@PadRight[{},4,n]],{1}]],{n,200}],PrimeQ] (* Harvey P. Dale, Oct 16 2017 *)

A210515 Numbers N such that concatenation of N, N, and x generates a prime for x=1 and x=3 and x=7 and x=9.

Original entry on oeis.org

1235, 4061, 8255, 22775, 24665, 36500, 44501, 52343, 54434, 57644, 58109, 59567, 59588, 65018, 69407, 71789, 78689, 94280, 98594, 106748, 114272, 122504, 134369, 137129, 138905, 144302, 162236, 196439, 235808, 238235, 269912, 277919, 278633, 282461, 290534

Offset: 1

Abhiram R Devesh, Jan 26 2013

The primes generated are part of the sequences A210511, A210512, A210513 and A210514.

There are no terms N with d = 3, 9, 15, 21, ... decimal digits since in those cases (10^(d+1)+10) is divisible by 7. - Michael S. Branicky, Apr 28 2025

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A210511, A210512, A210513, A210514.

Select[Range[3*10^5],AllTrue[FromDigits/@Table[Join[IntegerDigits[#],IntegerDigits [#],{n}],{n,{1,3,7,9}}],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 07 2021 *)

import numpy as np
from functools import reduce
def factors(n):
    return reduce(list._add_, ([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0))
for i in range(1,50000):
    p1=int(str(i)+str(i)+"1")
    p3=int(str(i)+str(i)+"3")
    p7=int(str(i)+str(i)+"7")
    p9=int(str(i)+str(i)+"9")
    if len(factors(p1))<3 and len(factors(p3))<3 and len(factors(p7))<3 and len(factors(p9))<3:
        print(i, end=',')

from gmpy2 import is_prime
def ok(n):
    b = n*(10**len(str(n))+1)*10
    return all(is_prime(b+x) for x in [1, 3, 7, 9])
print([N for N in range(3*10**5) if ok(N)]) # Michael S. Branicky, Apr 28 2025

A210534 Primes formed by concatenating palindromes having even number of digits with 1.

Original entry on oeis.org

331, 661, 881, 991, 12211, 14411, 15511, 20021, 21121, 23321, 24421, 29921, 33331, 35531, 41141, 45541, 47741, 50051, 51151, 57751, 59951, 63361, 71171, 72271, 74471, 75571, 81181, 84481, 99991, 1022011, 1255211, 1299211, 1311311, 1344311, 1355311

Offset: 1

Jonathan Vos Post, Jan 30 2013

Analogous to A210511, except that the second n is digit reversed. If the first (leftmost) n were reversed, we would have problems with trailing zeros becoming leading zeros, which get removed in OEIS formatting. That is a slightly different sequence is given by the formula primes of the form n concatenated with A004086(n) concatenated with "1"; or Primes of form a(n) = (n*10^A055642(n)+A004086(n)) concatenated with "1".

There are 190 terms up to all 6-digit palindromes (i.e., 7-digit primes), 1452 terms up to all 8-digit palindromes (i.e., 9-digit primes), and 11724 terms up to all 10-digit palindromes (i.e., 11-digit primes). - Harvey P. Dale, Jul 06 2018

			a(18) = 50 concatenated with R(50)=05 concatenated with "1" = 50051, which is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1452

Cf. A000040, A004086, A210511.

fulldigRev := proc(n)
    local digs ;
    digs := convert(n,base,10) ;
    [op(ListTools[Reverse](digs)),op(digs)] ;
end proc:
for n from 1 to 150 do
    r := [1,op(fulldigRev(n))] ;
    p := add(op(i,r)*10^(i-1),i=1..nops(r)) ;
    if isprime(p) then
        printf("%d,",p);
    end if;
end do: # R. J. Mathar, Feb 21 2013

10#+1&/@Select[Table[FromDigits[Join[IntegerDigits[n],Reverse[ IntegerDigits[ n]]]],{n,9999}],PrimeQ[10#+1]&](* Harvey P. Dale, Jul 06 2018 *)
10#+1&/@Select[Flatten[Table[Range[10^n,10^(n+1)],{n,1,5,2}]], PalindromeQ[ #] && PrimeQ[10#+1]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 11 2019 *)

A210711 Semiprimes formed by concatenating n, n, and 1 for n = 1, 2, 3,....

Original entry on oeis.org

111, 221, 551, 771, 11111, 14141, 15151, 16161, 19191, 23231, 24241, 29291, 30301, 34341, 36361, 37371, 38381, 39391, 42421, 44441, 47471, 50501, 53531, 55551, 56561, 59591, 62621, 68681, 70701, 74741, 75751, 77771, 79791, 81811, 83831, 84841, 87871, 91911, 95951, 96961, 1001001

Offset: 1

Jonathan Vos Post, Jan 29 2013

This is to A210511 as semiprimes A001358 are to primes A000040.

			a(1) = 11 because 111 = 3 * 37.
a(2) = 221 because 221 = 13 * 17.
331 is not in the sequence, because it is a prime.
a(5) = 11111 because "11" concatenated with "11" concatenated with "1" = 11111 = 41 * 271.

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

Cf. A001358, A210511.

IsSemiprime:=func;
[nn1: n in [1..100] | IsSemiprime(nn1) where nn1 is Seqint([1] cat Intseq(n) cat Intseq(n))]; // Bruno Berselli, Jan 30 2013

read("transforms"):
for n from 1 to 100 do
    L := [n,n,1] ;
    p := digcatL(L) ;
    if numtheory[bigomega](p) = 2 then
        print(p) ;
    end if;
end do: # R. J. Mathar, Jan 30 2013

A211401 a(n) = n-th prime of the form (k concatenated with k concatenated...) n times concatenated with 1.

Original entry on oeis.org

11, 661, 4441, 404040401, 29292929291, 5353535353531, 1291291291291291291291, 81818181818181811, 8888888888888888881, 2532532532532532532532532532531, 2282282282282282282282282282282281, 1201201201201201201201201201201201201, 3813813813813813813813813813813813813811

Offset: 1

Jonathan Vos Post, Feb 09 2013

Main diagonal A[n,n] of array A[k,n] = n-th prime of the form (j concatenated with j concatenated...) k times concatenated with 1.

			a(1) = 11 because that is the 1st (smallest) prime of the form Concatenate(1 copy of k) with 1, for k = 1, 2, 3, ....
a(2) = 661 because the 1st (smallest) prime of the form Concatenate(2 copies of k) with 1, for k = 1, 2, 3, .... is 331, and the 2nd is 661.
a(3) = 4441 because the 1st (smallest) prime of the form Concatenate(3 copies of k) with 1, for k = 1, 2, 3, .... is 2221, the 2nd is 3331, and the 3rd is 4441.
a(4) = 404040401 because the 1st (smallest) prime of the form Concatenate(4 copies of k) with 1, for k = 1, 2, 3, .... is 33331, the 2nd is 99991, the 3rd is 242424241, and the 4th is 404040401.

Cf. A030430, A210511, A210712, A210720.

A211401k := proc(n,k)
    option remember;
    local p,amin;
    if n = 1 then
        amin := 1 ;
    else
        amin := procname(n-1,k)+1 ;
    end if;
    for a from amin do
        [seq(a,i=1..k),1] ;
        p := digcatL(%) ;
        if isprime(p) then
            return a;
        end if;
    end do:
end proc:
A211401 := proc(n)
    b := A211401k(n,n) ;
    [seq(b,i=1..n),1] ;
    digcatL(%) ;
end proc: # R. J. Mathar, Feb 10 2013

Table[Select[Table[10 FromDigits[Flatten[IntegerDigits/@PadRight[{},k,n]]]+1,{n,1000}],PrimeQ][[k]],{k,15}] (* Harvey P. Dale, Oct 13 2022 *)

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A210712 Primes formed by concatenating n, n, n, and 1 for n = 1, 2, 3,....

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A210512 Primes formed by concatenating k, k and 3 for k >= 1.

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A210720 Primes formed by concatenating n, n, n, n, and 1 for n = 1, 2, 3,....

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A210515 Numbers N such that concatenation of N, N, and x generates a prime for x=1 and x=3 and x=7 and x=9.

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A210534 Primes formed by concatenating palindromes having even number of digits with 1.

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A210711 Semiprimes formed by concatenating n, n, and 1 for n = 1, 2, 3,....

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A211401 a(n) = n-th prime of the form (k concatenated with k concatenated...) n times concatenated with 1.

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