A262627 -id:A262627

A262628

Base-10 representation of 0 and the primes at A262627.

+20
2

0, 5, 1619, 22861, 5608661, 123097943, 2103004511, 411605194237, 29681120620523, 8356197277826743, 141511180168198879, 23016765965614427641, 30251327028834944270131, 371385619022538141977254297, 124784413675757116661951994259

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

OFFSET

1,2

LINKS

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

EXAMPLE

n A262627(n) base-10 representation

1 0 0

2 101 5

3 11001010011 1619

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. A262627. Subsequence of A016041 (except a(1)).

KEYWORD

nonn,base

AUTHOR

Clark Kimberling, Oct 02 2015

STATUS

approved

A262639

Minimal nested palindromic base-4 primes with seed 3; see Comments.

+10
4

3, 131, 11311, 121131121, 1212113112121, 312121131121213, 101312121131121213101, 11131013121211311212131013111, 31311131013121211311212131013111313, 1011313111310131212113112121310131113131101, 310113131113101312121131121213101311131311013

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

OFFSET

1,1

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 palindromic base-4 primes with seed s.

LINKS

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

EXAMPLE

a(3) = 11311 is the least base-4 prime having a(2) = 131 in its middle.

Triangular format:

3

131

11311

121131121

1212113112121

312121131121213

MATHEMATICA

s = {3}; 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 (* A262639 *)

Map[FromDigits[ToString[#], base] &, s] (* A262640 *)

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

CROSSREFS

Cf. A261881 (base 10), A262640, A262627.

KEYWORD

nonn,easy,base

AUTHOR

Clark Kimberling, Oct 24 2015

STATUS

approved

A262645

Minimal nested palindromic base-6 primes with seed 0; see Comments.

+10
4

0, 101, 5110115, 13511011531, 1135110115311, 111351101153111, 152111351101153111251, 5215211135110115311125125, 1025215211135110115311125125201, 1431025215211135110115311125125201341, 1111431025215211135110115311125125201341111

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

OFFSET

1,2

COMMENTS

Using only base-6 digits 0,1,2,3,4,5, 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 palindromic base-6 primes with seed s.

LINKS

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

EXAMPLE

a(3) = 5110115 is the least base-6 prime having a(2) = 101 in its middle.

Triangular format:

0

101

5110115

13511011531

1135110115311

111351101153111

MATHEMATICA

s = {0}; base = 6; 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 (* A262645 *)

Map[FromDigits[ToString[#], base] &, s] (* A262646 *)

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

CROSSREFS

Cf. A261881 (base 10), A262646, A262627.

KEYWORD

nonn,base

AUTHOR

Clark Kimberling, Oct 24 2015

STATUS

approved

A262651

Minimal nested palindromic base-6 primes with seed 3; see Comments.

+10
4

3, 11311, 121131121, 5312113112135, 14531211311213541, 1145312113112135411, 51114531211311213541115, 5511145312113112135411155, 50551114531211311213541115505, 115055111453121131121354111550511, 51150551114531211311213541115505115

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

OFFSET

1,1

COMMENTS

Using only base-6 digits 0,1,2,3,4,5, 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 palindromic base-6 primes with seed s.

LINKS

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

EXAMPLE

a(3) = 121131121 is the least base-6 prime having a(2) = 11311 in its middle. Triangular format:

3

11311

121131121

5312113112135

14531211311213541

MATHEMATICA

s = {3}; base = 6; 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 (* A262651 *)

Map[FromDigits[ToString[#], base] &, s] (* A262652 *)

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

CROSSREFS

Cf. A261881 (base 10), A262652, A262627.

KEYWORD

nonn,easy,base

AUTHOR

Clark Kimberling, Oct 27 2015

STATUS

approved

A262629

Minimal nested base-2 palindromic primes with seed 1.

+10
3

1, 111, 11111, 1111111, 1001111111001, 1001001111111001001, 111110010011111110010011111, 111111110010011111110010011111111, 100111111110010011111110010011111111001, 1011010011111111001001111111001001111111100101101

(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.

LINKS

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

EXAMPLE

a(5) = 1001111111001 = A117697(20) is the least base-2 prime having a(4) = 1111111 = A117697(8) in its middle. Triangular format:

1

111

11111

1111111

1001111111001

1001001111111001001

111110010011111110010011111

MATHEMATICA

s = {1}; 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 (* A262629 *)

Map[FromDigits[ToString[#], base] &, s] (* A262630 *)

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

CROSSREFS

Cf. A261881 (base 10), A262627. Subsequence of A117697 (expect a(1)).

KEYWORD

nonn,base

AUTHOR

Clark Kimberling, Oct 02 2015

STATUS

approved

A262631

Minimal nested base-3 palindromic primes with seed 1.

+10
3

1, 111, 1111111, 22111111122, 1221111111221, 112211111112211, 2111221111111221112, 2102111221111111221112012, 1212102111221111111221112012121, 20121210211122111111122111201212102, 2002201212102111221111111221112012121022002

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

OFFSET

1,2

COMMENTS

Using only base-3 digits 0,1,2, 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-3 palindromic primes with seed s.

LINKS

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

EXAMPLE

a(4) = 22111111122 is the least base-3 prime having a(3) = 1111111 in its middle. Triangular format:

1

111

1111111

22111111122

1221111111221

112211111112211

MATHEMATICA

s = {1}; base = 3; 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 (* A262631 *)

Map[FromDigits[ToString[#], base] &, s] (* A262632 *)

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

CROSSREFS

Cf. A261881 (base 10), A262632, A262627. Subset of A117698 (except a(1)).

KEYWORD

nonn,base

AUTHOR

Clark Kimberling, Oct 02 2015

STATUS

approved

A262633

Minimal nested base-4 palindromic primes with seed 0.

+10
3

0, 101, 31013, 3310133, 1023310133201, 3331023310133201333, 3223331023310133201333223, 1133223331023310133201333223311, 100311332233310233101332013332233113001, 10231003113322333102331013320133322331130013201

(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) = 31013 is the least base-4 prime having a(2) = 101 in its middle. Triangular format:

0

101

31013

3310133

1023310133201

3331023310133201333,

MATHEMATICA

s = {0}; 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 (* A262633 *)

Map[FromDigits[ToString[#], base] &, s] (* A262634 *)

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

CROSSREFS

Cf. A261881 (base 10), A262634, A262627. Subsequence of A117699.

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:

1

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 *)

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

CROSSREFS

Cf. A261881 (base 10), A262636, A262627. Subsequence of A117699.

KEYWORD

nonn,base

AUTHOR

Clark Kimberling, Oct 02 2015

STATUS

approved

A262637

Minimal nested palindromic base-4 primes with seed 2; see Comments.

+10
3

2, 323, 3332333, 333323333, 33333233333, 103333323333301, 1210333332333330121, 100212103333323333301212001, 3310021210333332333330121200133, 3303310021210333332333330121200133033, 11330331002121033333233333012120013303311

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

OFFSET

1,1

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 palindromic base-4 primes with seed s.

LINKS

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

EXAMPLE

a(3) = 3332333 is the least base-4 prime having a(2) = 323 in its middle.

Triangular format:

2

323

3332333

333323333

33333233333

103333323333301

1210333332333330121

MATHEMATICA

s = {2}; 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 (* A262637 *)

Map[FromDigits[ToString[#], base] &, s] (* A262638 *)

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

CROSSREFS

Cf. A261881 (base 10), A262638, A262627.

KEYWORD

nonn,easy,base

AUTHOR

Clark Kimberling, Oct 24 2015

STATUS

approved

A262641

Minimal nested palindromic base-5 primes with seed 1; see Comments.

+10
3

1, 111, 41114, 1411141, 20141114102, 12120141114102121, 24012120141114102121042, 142401212014111410212104241, 41424012120141114102121042414, 314142401212014111410212104241413, 4131414240121201411141021210424141314, 30413141424012120141114102121042414131403

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

OFFSET

1,2

COMMENTS

Using only base-5 digits 0,1,2,3,4, 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 palindromic base-5 primes with seed s.

LINKS

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

EXAMPLE

a(3) = 41114 is the least base-5 prime having a(2) = 111 in its middle.

Triangular format:

1

111

41114

1411141

20141114102

12120141114102121

24012120141114102121042

MATHEMATICA

s = {1}; base = 5; 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 (* A262641 *)

Map[FromDigits[ToString[#], base] &, s] (* A262642 *)

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

CROSSREFS

Cf. A261881 (base 10), A262627, A262642.

KEYWORD

nonn,easy,base

AUTHOR

Clark Kimberling, Oct 24 2015

STATUS

approved