A261881 -id:A261881 - cp's OEIS frontend

A262627 Minimal nested base-2 palindromic primes with seed 0.

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

Offset: 1

Clark Kimberling, Oct 02 2015

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.
       0      A262627        A262628
       1      A262629        A262630
       1      A262631        A262632
       0      A262633        A262634
       1      A262635        A262636
       2      A262637        A262638
       3      A262639        A262640
       1      A262641        A262642
       3      A262643        A262644
       0      A262645        A262646
       1      A262647        A262648
       2      A262649        A262650
       3      A262651        A262652
       4      A262653        A262654
       5      A262655        A262656
       1      A262657        A262658
       0      A262659        A262660
       1      A262661        A262662
      0      A261881        A261881
      1      A261818        A261818

			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

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

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

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 *)

A053600 a(1) = 2; for n>=1, a(n+1) is the smallest palindromic prime with a(n) as a central substring.

Original entry on oeis.org

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

Offset: 1

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

			As a triangle:
.........2
........727
.......37273
.....333727333
....93337273339
..309333727333903
1830933372733390381

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

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.

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

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 *)

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

A082563 a(1) = 3; for n>=1, a(n+1) is the smallest palindromic prime with a(n) as a central substring.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, May 04 2003

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

			As a triangle:
........3
.......131
......11311
....121131121
..1212113112121
36121211311212163

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

Cf. A053600, A075597, A261881.

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 *)

Name changed by Arkadiusz Wesolowski, Sep 15 2011

More terms from Clark Kimberling, Sep 23 2015

A262639 Minimal nested palindromic base-4 primes with seed 3; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 24 2015

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.

			a(3) = 11311 is the least base-4 prime having a(2) = 131 in its middle.
Triangular format:
         3
        131
       11311
     121131121
   1212113112121
  312121131121213

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

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

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 *)

A262645 Minimal nested palindromic base-6 primes with seed 0; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 24 2015

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.

			a(3) = 5110115 is the least base-6 prime having a(2) = 101 in its middle.
Triangular format:
         0
        101
      5110115
    13511011531
   1135110115311
  111351101153111

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

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

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 *)

A262651 Minimal nested palindromic base-6 primes with seed 3; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 27 2015

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.

			a(3) = 121131121 is the least base-6 prime having a(2) = 11311 in its middle. Triangular format:
        3
      11311
    121131121
  5312113112135
14531211311213541

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

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

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 *)

A261818 Minimal nested palindromic primes with seed 1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 17 2015

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.

			As a symmetric triangle:
......1
.....313
....93139
...3931393
.11393139311
1113931393111

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

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

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 *)

A262629 Minimal nested base-2 palindromic primes with seed 1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 02 2015

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(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

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

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

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 *)

A262631 Minimal nested base-3 palindromic primes with seed 1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 02 2015

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.

			a(4) = 22111111122 is the least base-3 prime having a(3) = 1111111 in its middle. Triangular format:
         1
        111
      1111111
    22111111122
   1221111111221
  112211111112211

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

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

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 *)

A262633 Minimal nested base-4 palindromic primes with seed 0.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 02 2015

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.

			a(3) = 31013 is the least base-4 prime having a(2) = 101 in its middle. Triangular format:
         0
        101
       31013
      3310133
   1023310133201
3331023310133201333,

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

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

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 *)

cp's OEIS Frontend

A262627 Minimal nested base-2 palindromic primes with seed 0.

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A053600 a(1) = 2; for n>=1, a(n+1) is the smallest palindromic prime with a(n) as a central substring.

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A082563 a(1) = 3; for n>=1, a(n+1) is the smallest palindromic prime with a(n) as a central substring.

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A262639 Minimal nested palindromic base-4 primes with seed 3; see Comments.

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A262645 Minimal nested palindromic base-6 primes with seed 0; see Comments.

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A262651 Minimal nested palindromic base-6 primes with seed 3; see Comments.

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A261818 Minimal nested palindromic primes with seed 1.

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A262629 Minimal nested base-2 palindromic primes with seed 1.

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