A261881 -id:A261881

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:

0

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

A053600

a(1) = 2; for n>=1, a(n+1) is the smallest palindromic prime with a(n) as a central substring.

+10
13

2, 727, 37273, 333727333, 93337273339, 309333727333903, 1830933372733390381, 92183093337273339038129, 3921830933372733390381293, 1333921830933372733390381293331, 18133392183093337273339038129333181

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

OFFSET

1,1

REFERENCES

G. L. Honaker, Jr. and Chris K. Caldwell, Palindromic Prime Pyramids, J. Recreational Mathematics, Vol. 30(3) 169-176, 1999-2000.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..200

P. De Geest, Palindromic Prime Pyramid Puzzle by G.L.Honaker,Jr

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 18133...33181 (35-digits)

G. L. Honaker, Jr. & C. K. Caldwell, Palindromic Prime Pyramids

G. L. Honaker, Jr. & C. K. Caldwell, Supplement to "Palindromic Prime Pyramids"

Ivars Peterson, Primes, Palindromes, and Pyramids, Science News.

Inder J. Taneja, Palindromic Prime Embedded Trees, RGMIA Res. Rep. Coll. 20 (2017), Art. 124.

Inder J. Taneja, Same Digits Embedded Palprimes, RGMIA Research Report Collection (2018) Vol. 21, Article 75, 1-47.

EXAMPLE

As a triangle:

.........2

........727

.......37273

.....333727333

....93337273339

..309333727333903

1830933372733390381

MATHEMATICA

d[n_] := IntegerDigits[n]; t = {x = 2}; Do[i = 1; While[! PrimeQ[y = FromDigits[Flatten[{z = d[i], d[x], Reverse[z]}]]], i++]; AppendTo[t, x = y], {n, 10}]; t (* Jayanta Basu, Jun 24 2013 *)

PROG

(Python)

from gmpy2 import digits, mpz, is_prime

A053600_list, p = [2], 2

for _ in range(30):

m, ps = 1, digits(p)

s = mpz('1'+ps+'1')

while not is_prime(s):

m += 1

ms = digits(m)

s = mpz(ms+ps+ms[::-1])

p = s

A053600_list.append(int(p)) # Chai Wah Wu, Apr 09 2015

CROSSREFS

Cf. A000040, A002385, A047076, A052205, A034276, A256957, A052091, A052092, A261881.

KEYWORD

base,nonn

AUTHOR

G. L. Honaker, Jr., Jan 20 2000

STATUS

approved

A082563

a(1) = 3; for n>=1, a(n+1) is the smallest palindromic prime with a(n) as a central substring.

+10
4

3, 131, 11311, 121131121, 1212113112121, 36121211311212163, 303612121131121216303, 7230361212113112121630327, 30723036121211311212163032703, 723072303612121131121216303270327, 1472307230361212113112121630327032741, 114723072303612121131121216303270327411

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

OFFSET

1,1

COMMENTS

The minimal nested palindromic primes with seed 3; see A261881 for a guide to related sequences.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..200

EXAMPLE

As a triangle:

........3

.......131

......11311

....121131121

..1212113112121

36121211311212163

MATHEMATICA

s = {3}; Do[NestWhile[# + 1 &, 1, ! PrimeQ[tmp = FromDigits[Join[#, IntegerDigits[Last[s]], Reverse[#]] &[IntegerDigits[#]]]] &]; AppendTo[s, tmp], {15}]; s

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

CROSSREFS

Cf. A053600, A075597, A261881.

KEYWORD

nonn,base

AUTHOR

Benoit Cloitre, May 04 2003

EXTENSIONS

Name changed by Arkadiusz Wesolowski, Sep 15 2011

More terms from Clark Kimberling, Sep 23 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

A261818

Minimal nested palindromic primes with seed 1.

+10
3

1, 313, 93139, 3931393, 11393139311, 1113931393111, 17111393139311171, 331711139313931117133, 3333171113931393111713333, 133331711139313931117133331, 1813333171113931393111713333181, 1951813333171113931393111713333181591

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

OFFSET

1,2

COMMENTS

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 primes with seed s.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..100

EXAMPLE

As a symmetric triangle:

......1

.....313

....93139

...3931393

.11393139311

1113931393111

MATHEMATICA

s = {1}; Do[NestWhile[# + 1 &, 1, ! PrimeQ[tmp = FromDigits[Join[#, IntegerDigits[Last[s]], Reverse[#]] &[IntegerDigits[#]]]] &]; AppendTo[s, tmp], {15}]; s

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

CROSSREFS

Cf. A261881 (seed 0 with guide to related sequences).

KEYWORD

nonn,easy,base

AUTHOR

Clark Kimberling, Sep 17 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