A088281 -id:A088281 - cp's OEIS frontend

A332118 a(n) = (10^(2n+1) - 1)/9 + 7*10^n.

8, 181, 11811, 1118111, 111181111, 11111811111, 1111118111111, 111111181111111, 11111111811111111, 1111111118111111111, 111111111181111111111, 11111111111811111111111, 1111111111118111111111111, 111111111111181111111111111, 11111111111111811111111111111, 1111111111111118111111111111111

Offset: 0

M. F. Hasler, Feb 09 2020

See A107648 = {1, 4, 6, 7, 384, 666, ...} for the indices of primes.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Patrick De Geest, Palindromic Wing Primes: (1)8(1), updated: June 25, 2017.
Makoto Kamada, Factorization of 11...11811...11, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077791-1)/2 = A107648: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes), A077798 (palindromic wing primes), A088281 (primes 1..1x1..1), A068160 (smallest of given length), A053701 (vertically symmetric numbers).
Cf. A332128 .. A332178, A181965 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

A332118 := n -> (10^(2*n+1)-1)/9+7*10^n;

Array[(10^(2 # + 1)-1)/9 + 7*10^# &, 15, 0]

apply( {A332118(n)=10^(n*2+1)\9+7*10^n}, [0..15])

def A332118(n): return 10**(n*2+1)//9+7*10**n

a(n) = A138148(n) + 8*10^n = A002275(2n+1) + 7*10^n.
G.f.: (8 - 707*x + 600*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A088282 Palindromic primes in which a single digit is sandwiched between strings of 3's.

Original entry on oeis.org

313, 353, 373, 383, 33533, 3331333, 3337333, 333333313333333, 333333373333333, 333333383333333, 33333333333733333333333, 333333333333373333333333333, 33333333333333333533333333333333333, 33333333333333333733333333333333333

Offset: 1

Amarnath Murthy, Sep 29 2003

a(36) has 1553 digits and is therefore too large to include in the b-file. - Harvey P. Dale, Mar 22 2020

Harvey P. Dale, Table of n, a(n) for n = 1..35

Cf. A088281, A088283, A088284.

Select[FromDigits/@Flatten[Table[Join[PadRight[{},k,3],{n},PadRight[ {},k,3]],{n,0,9},{k,20}],1],PrimeQ]//Sort (* Harvey P. Dale, Mar 22 2020 *)

More terms from David Wasserman, Aug 03 2005

A088283 Palindromic primes in which a single digit is sandwiched between strings of 7's.

Original entry on oeis.org

727, 757, 787, 797, 77377, 77477, 77977, 7772777, 7774777, 7778777, 777767777, 77777677777, 7777774777777, 777777727777777, 777777757777777, 77777777677777777, 77777777977777777, 777777777727777777777

Offset: 1

Amarnath Murthy, Sep 29 2003

Harvey P. Dale, Table of n, a(n) for n = 1..37

Cf. A088281, A088282, A088284.

Select[Flatten[Table[FromDigits[Flatten[Join[{PadRight[{},n,7], i, PadRight[ {},n,7]}]]],{n,10},{i,0,9}]],PrimeQ] (* Harvey P. Dale, Jun 15 2015 *)

More terms from David Wasserman, Aug 03 2005

A088284 Palindromic primes in which a single digit is sandwiched between nonempty strings of 9's.

Original entry on oeis.org

919, 929, 99999199999, 99999999299999999, 9999999992999999999, 999999999999919999999999999, 99999999999999499999999999999, 999999999999999999999949999999999999999999999, 99999999999999999999999999899999999999999999999999999

Offset: 1

Amarnath Murthy, Sep 29 2003

Alois P. Heinz, Table of n, a(n) for n = 1..20
Hans Havermann, Near-rep-nine palindromic primes

Subsequence of A002385.

Cf. A088281, A088282, A088283.

Select[Flatten[Table[FromDigits[Join[PadRight[{},n,9],{d},PadRight[{},n,9]]],{n,30},{d,Range[8]}]],PrimeQ] (* Harvey P. Dale, Mar 17 2023 *)

More terms from David Wasserman, Aug 03 2005

Name clarified by Christian Stump, Mar 31 2015

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.

Original entry on oeis.org

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

A046705 Palindromic primes whose product of digits is a prime.

Original entry on oeis.org

2, 3, 5, 7, 131, 151, 11311, 1117111, 111111151111111, 111111111111111111131111111111111111111, 1111111111111111111111111111111117111111111111111111111111111111111, 1111111111111111111111111111111111111111111115111111111111111111111111111111111111111111111

Offset: 1

Felice Russo

Except for the first 4 terms, a subsequence of A088281. - Chai Wah Wu, Dec 17 2015

Subsequence of A028842, of A046703, and also of A117058. - Michel Marcus, Dec 18 2015

Chai Wah Wu, Table of n, a(n) for n = 1..15

Cf. A028842, A046703, A088281, A117058.

t = Prime[Range[4]]; Union[Select[Flatten[Table[NestList[FromDigits[Flatten[{1, IntegerDigits[#], 1}]] &, n, 45], {n, t}]], PrimeQ]] (* Jayanta Basu, Jun 27 2013 *)

from _future_ import division
from sympy import isprime
A046705_list = [n for n in ((10**(2*l+1)-1)//9+d*10**l for l in range(100) for d in [1,2,4,6]) if isprime(n)] # Chai Wah Wu, Dec 17 2015

A345223 a(n) is the smallest k >= 0 such that the decimal concatenation 1 (n times) || k || 1 (n times) is a prime, or -1 if no such k exists.

Original entry on oeis.org

0, 3, 4, 8, 10, 8, 5, 21, 1, 6, 1, 116, 23, 6, 73, 24, 16, 62, 3, 10, 19, 53, 61, 58, 191, 9, 265, 12, 133, 86, 141, 4, 7, 39, 193, 31, 51, 13, 31, 6, 31, 53, 287, 139, 4, 239, 187, 25, 18, 144, 31, 38, 93, 86, 27, 30, 16, 24, 6, 356, 50, 91, 395, 117, 217, 61

Offset: 1

Felix Fröhlich, Jun 11 2021

a(n) = 0 only for n = 1, since A138148(1) = 101 is the only prime in A138148.
a(n) = 1 iff n is of the form (A004023(i)-1)/2 for some i >= 1.
No term equals 2, see second comment in A258372.

			For n = 3: 1110111, 1111111, 1112111 and 1113111 are all composite, while 1114111 is prime, so the smallest number that can be inserted between strings of three ones so that the concatenation is prime is 4. Therefore a(3) = 4.

Cf. A004023, A088281, A138148, A258372.

Table[Module[{k=0},While[!PrimeQ[FromDigits[Flatten[Join[{PadRight[ {},n,1],IntegerDigits[ k],PadRight[{},n,1]}]]]],k++];k],{n,70}] (* Harvey P. Dale, Jun 03 2024 *)

eva(n) = subst(Pol(n), x, 10)
a(n) = my(v=vector(n, t, 1), d, w=[]); for(k=0, oo, d=digits(k); w=concat(v, d); w=concat(w, v); if(ispseudoprime(eva(w)), return(k)))

from sympy import isprime
def a(n, d=1):
    k, bread = 0, str(d)*n
    while not isprime(int(bread + str(k) + bread)): k += 1
    return k
print([a(n) for n in range(1, 67)]) # Michael S. Branicky, Jun 11 2021

cp's OEIS Frontend

A332118 a(n) = (10^(2n+1) - 1)/9 + 7*10^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A088282 Palindromic primes in which a single digit is sandwiched between strings of 3's.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A088283 Palindromic primes in which a single digit is sandwiched between strings of 7's.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A088284 Palindromic primes in which a single digit is sandwiched between nonempty strings of 9's.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A046705 Palindromic primes whose product of digits is a prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

A345223 a(n) is the smallest k >= 0 such that the decimal concatenation 1 (n times) || k || 1 (n times) is a prime, or -1 if no such k exists.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python