A262636 -id:A262636

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A262627

Minimal nested base-2 palindromic primes with seed 0.

+10
19

0, 101, 11001010011, 101100101001101, 10101011001010011010101, 111010101100101001101010111, 1111101010110010100110101011111, 101111111010101100101001101010111111101, 110101111111010101100101001101010111111101011

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Using only base-2 digits 0 and 1, let s be a palindrome and put a(1) = s. Let a(2) be the least palindromic prime having s in the middle; for n > 2, let a(n) be the least palindromic prime have a(n-1) in the middle. Then (a(n)) is the sequence of minimal nested base-2 palindromic primes with seed s -- a(1) being not prime, of course.

Guide to related sequences

base seed base-b repr. base-10 repr.

2 0 A262627 A262628

2 1 A262629 A262630

3 1 A262631 A262632

4 0 A262633 A262634

4 1 A262635 A262636

4 2 A262637 A262638

4 3 A262639 A262640

5 1 A262641 A262642

5 3 A262643 A262644

6 0 A262645 A262646

6 1 A262647 A262648

6 2 A262649 A262650

6 3 A262651 A262652

6 4 A262653 A262654

6 5 A262655 A262656

7 1 A262657 A262658

8 0 A262659 A262660

8 1 A262661 A262662

10 0 A261881 A261881

10 1 A261818 A261818

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..300

EXAMPLE

a(3) = 11001010011 =A117697(15) is the least prime having a(2) = 101 in its middle. Triangular format:

101

11001010011

101100101001101

10101011001010011010101

111010101100101001101010111

1111101010110010100110101011111

MATHEMATICA

s = {0}; base = 2; z = 20; Do[NestWhile[# + 1 &, 1, ! PrimeQ[tmp = FromDigits[Join[#, IntegerDigits[Last[s]], Reverse[#]] &[IntegerDigits[#, base]], base]] &];

AppendTo[s, FromDigits[IntegerDigits[tmp, base]]], {z}]; s (* A262627 *)

Map[FromDigits[ToString[#], base] &, s] (* A262628 *)

(* Peter J. C. Moses, Sep 01 2015 *)

CROSSREFS

Cf. A117697, A261881 (base 10), A262628-A262662.

KEYWORD

nonn,base

AUTHOR

Clark Kimberling, Oct 02 2015

STATUS

approved

A262635

Minimal nested base-4 palindromic primes with seed 1.

+10
3

1, 12121, 111212111, 31112121113, 133111212111331, 123133111212111331321, 303123133111212111331321303, 3030312313311121211133132130303, 30303031231331112121113313213030303, 3303030312313311121211133132130303033, 11330303031231331112121113313213030303311

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Using only base-4 digits 0,1,2,3, let s be a palindrome and put a(1) = s. Let a(2) be the least palindromic prime having s in the middle; for n > 2, let a(n) be the least palindromic prime have a(n-1) in the middle. Then (a(n)) is the sequence of minimal nested base-4 palindromic primes with seed s.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..300

EXAMPLE

a(3) = 111212111 is the least base-4 prime having a(2) = 12121 in its middle. Triangular format:

12121

111212111

31112121113

133111212111331

123133111212111331321

MATHEMATICA

s = {1}; base = 4; z = 20; Do[NestWhile[# + 1 &, 1, ! PrimeQ[tmp = FromDigits[Join[#, IntegerDigits[Last[s]], Reverse[#]] &[IntegerDigits[#, base]], base]] &];

AppendTo[s, FromDigits[IntegerDigits[tmp, base]]], {z}]; s (* A262635 *)

Map[FromDigits[ToString[#], base] &, s] (* A262636 *)