A053600 -id:A053600 - cp's OEIS frontend

A052092 Lengths of the palindromic primes from Honaker's sequence A053600.

1, 3, 5, 9, 11, 15, 19, 23, 25, 31, 35, 41, 45, 49, 55, 59, 63, 69, 75, 81, 87, 93, 99, 105, 109, 113, 119, 125, 129, 133, 139, 145, 151, 157, 161, 167, 173, 179, 185, 191, 195, 201, 207, 213, 219, 225, 231, 237, 243, 249, 255, 261, 267, 273, 279, 285, 291, 297

Offset: 0

Patrick De Geest, Jan 15 2000

Since the terms from a(34) onward are currently only probable primes, the lengths given in this sequence beyond that point are only provisional.

For n > 0, a(n) = a(n-1)+2*m where m is the number of digits of A052091(n). - Chai Wah Wu, Dec 03 2015

Chai Wah Wu, Table of n, a(n) for n = 0..501

Cf. A052091, A053600, A047076.

from sympy import isprime
A052092_list, l, p = [1], 1, 2
for _ in range(100):
    m, ps = 1, str(p)
    s = int('1'+ps+'1')
    while not isprime(s):
        m += 1
        ms = str(m)
        if ms[0] in '268':
            ms = str(int(ms[0])+1) + '0'*(len(ms)-1)
            m = int(ms)
        if ms[0] in '45':
            ms = '7' + '0'*(len(ms)-1)
            m = int(ms)
        s = int(ms+ps+ms[::-1])
    p = s
    l += 2*len(ms)
    A052092_list.append(l) # Chai Wah Wu, Dec 02 2015

Comments from G. L. Honaker, Jr., Mar 30 2000

Edited by T. D. Noe, Oct 30 2008

A261881 Minimal nested palindromic primes with seed 0.

Original entry on oeis.org

0, 101, 31013, 3310133, 933101339, 1093310133901, 30109331013390103, 333010933101339010333, 33330109331013390103333, 993333010933101339010333399, 104993333010933101339010333399401, 7810499333301093310133901033339940187

Offset: 1

Clark Kimberling, Sep 23 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 having a(n-1) in the middle.  Then (a(n)) is the sequence of minimal nested palindromic primes with seed s.
Guide to related sequences:
seed  sequence
0     A261881
1     A261818
2     A053600
3     A082563
4     A261821
5     A261822
6     A261823
7     A261824
8     A261825
9     A261826
000   A262531
0^5   A262532
0^7   A262533
0^9   A260250
010   A260459
111   A261820
1^5   A262497
1^7   A262498
1^9   A262499

			As a triangle:
........0
.......101
......31013
.....3310133
....933101339
..1093310133901
30109331013390103

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

Cf. A261818.

s = {0}; 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 *)

A047076 a(n+1) is the smallest palindromic prime containing exactly 2 more digits on each end than the previous term, with a(n) as a central substring.

Original entry on oeis.org

2, 30203, 133020331, 1713302033171, 12171330203317121, 151217133020331712151, 1815121713302033171215181, 16181512171330203317121518161, 331618151217133020331712151816133, 9333161815121713302033171215181613339, 11933316181512171330203317121518161333911

Offset: 1

G. L. Honaker, Jr., Jan 26 2000

Pickover, Clifford A., A Passion for Mathematics: Numbers, Puzzles, Madness, Religion and the Quest for Reality, John Wiley & Sons, Inc., Hoboken, New Jersey (2005) p. 108.

C. K. Caldwell, Palindromic Primes

Cf. A053600.

cc=If[EvenQ[First[#]],Reverse[#],#]&/@Tuples[{Range[0,9],Range[1,9,2]}]; nxt[n_]:=First[Select[Sort[FromDigits[Flatten[Join[{#,IntegerDigits[ n], Reverse[#]}]]]&/@cc],PrimeQ]]; NestList[nxt,2,10] (* Harvey P. Dale, Dec 24 2012 *)

A052205 a(n+1) is smallest palindromic prime containing exactly 3 more digits on each end than the previous term, with a(n) as a central substring.

Original entry on oeis.org

2, 1022201, 1051022201501, 1241051022201501421, 1071241051022201501421701, 1051071241051022201501421701501, 1091051071241051022201501421701501901, 1351091051071241051022201501421701501901531

Offset: 1

G. L. Honaker, Jr., Jan 28 2000

Giovanni Resta, Table of n, a(n) for n = 1..26 (full sequence)

Cf. A047076, A053600, A034276, A256957, A052091, A052092.

Shown to be finite by Felice Russo

A052091 Left parts needed for the construction of the palindromic prime pyramid starting with 2.

Original entry on oeis.org

2, 7, 3, 33, 9, 30, 18, 92, 3, 133, 18, 117, 17, 15, 346, 93, 33, 180, 120, 194, 126, 336, 331, 330, 95, 12, 118, 369, 39, 32, 165, 313, 165, 134, 13, 149, 195, 145, 158, 720, 18, 396, 193, 102, 737, 964, 722, 156, 106, 395, 945, 303, 310, 113, 150, 303, 715, 123

Offset: 0

Patrick De Geest, Jan 15 2000

Each term is the smallest to have the previous term as a centered substring, beginning with the smallest palindromic prime 2. The right parts are the reversals of the above terms leading zeros included. The terms from a(34) onward currently correspond only to strong pseudoprimes.

For n > 0, the leftmost (most significant) digit of a(n) is either 1, 3, 7 or 9. - Chai Wah Wu, Dec 02 2015

			Start with 2; add 7 gives 727; add 3 gives 37273; add 33 gives 333727333; etc.

Chai Wah Wu, Table of n, a(n) for n = 0..501
P. De Geest, World!Of Palindromic Primes, Page 3

Cf. A053600, A052092, A047076.

from sympy import isprime
A052091_list, p = [2], 2
for _ in range(30):
    m, ps = 1, str(p)
    s = int('1'+ps+'1')
    while not isprime(s):
        m += 1
        ms = str(m)
        if ms[0] in '268':
            ms = str(int(ms[0])+1) + '0'*(len(ms)-1)
            m = int(ms)
        if ms[0] in '45':
            ms = '7' + '0'*(len(ms)-1)
            m = int(ms)
        s = int(ms+ps+ms[::-1])
    p = s
    A052091_list.append(m) # Chai Wah Wu, Dec 02 2015

Comments from G. L. Honaker, Jr., Mar 30 2000

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

A256957 Smallest palindromic prime that generates a palindromic prime pyramid of height n.

Original entry on oeis.org

11, 131, 2, 5, 10301, 16361, 10281118201, 35605550653, 7159123219517, 17401539893510471, 3205657651567565023, 14736384418081448363741

Offset: 1

Felice Russo, Jan 25 2000

Start with a palindromic prime p; look for smallest palindromic prime that has previous term as a centered substring and has 2 more digits (i.e., one more digit at each end); repeat until no such palindromic prime can be found; then height(p) = number of rows in pyramid. Each row of pyramid must be the smallest prime that can be used. Then a(n) = smallest value of p that generates a pyramid of height n.

			a(1) = 11.
a(4) = 5:
5
151
31513
3315133, stop;
height(5)=4.
a(6)=16362:
16361
1163611
311636113
33116361133
3331163611333
333311636113333, stop;
height(16361)=6.

G. L. Honaker, Jr. and Chris K. Caldwell, Palindromic prime pyramids
Ivars Peterson's MathTrek, Primes, Palindromes, and Pyramids
Chai Wah Wu, On a conjecture regarding primality of numbers constructed from prepending and appending identical digits, arXiv:1503.08883 [math.NT], 2015.

Cf. A034276, A052205, A053600.

Added a(10)-a(11) and corrected a(4) -  Chai Wah Wu, Apr 09 2015
Entry revised by N. J. A. Sloane, Apr 13 2015
a(12) from Michael S. Branicky, Oct 28 2024

A034276 Smallest prime that generates a prime pyramid of height n.

Original entry on oeis.org

11, 29, 2, 5, 41, 251, 43, 145577, 51941, 4372877, 26901631, 366636187, 15387286403, 218761753811, 3313980408469

Offset: 1

Felice Russo, Jan 25 2000

Let p be prime; look for the smallest prime in {1|p|1, 3|p|3, 7|p|7, 9|p|9}, where '|' stands for concatenation; repeat until no such prime can be found; then height(p) = number of rows in pyramid.

a(13) > 10^10. - Donovan Johnson, Aug 13 2010

			Example for p=43: 43 3433 334333 93343339 3933433393 939334333939 39393343339393, stop; height(43)=7.

H. Heinz, Patterns in Primes, Illustrating Two Prime Pyramids
Chai Wah Wu, On a conjecture regarding primality of numbers constructed from prepending and appending identical digits, arXiv:1503.08883 [math.NT], 2015.

Cf. A047076, A053600, A052205, A256957, A052091, A052092.

More terms from Naohiro Nomoto, Jul 14 2001
a(11)-a(12) from Donovan Johnson, Aug 13 2010
a(13) from Chai Wah Wu, Apr 10 2015
a(14)-a(15) from Giovanni Resta, May 15 2020

A247483 a(1)=4; for n>=1, a(n+1) is the smallest palindromic semiprime with a(n) as a central substring.

Original entry on oeis.org

4, 141, 41414, 1414141, 314141413, 33141414133, 3331414141333, 14333141414133341, 121433314141413334121, 1012143331414141333412101, 17101214333141414133341210171, 3171012143331414141333412101713, 19317101214333141414133341210171391

Offset: 1

Michel Lagneau, Dec 01 2014

			a(1) = 4 = 2*2;
a(2) = 141 = 3*47;
a(3) = 41414 = 2*20707;
a(4) = 1414141 = 43*32887.

Cf. A001358, A053600.

d[n_] := IntegerDigits[n]; t = {x = 4}; Do[i = 1; While[!PrimeOmega[y = FromDigits[Flatten[{z = d[i], d[x], Reverse[z]}]]]==2, i++]; AppendTo[t, x = y], {n, 12}]; t
nxt[n_]:=Module[{k=1},While[PrimeOmega[FromDigits[Join[IntegerDigits[k],IntegerDigits[n],Reverse[IntegerDigits[k]]]]]!=2,k++]; FromDigits[ Join[IntegerDigits[k],IntegerDigits[n],Reverse[IntegerDigits[k]]]]]; NestList[nxt,4,15] (* Harvey P. Dale, Oct 30 2024 *)

A375690 a(1) = 3; for n > 1, a(n) is the smallest palindromic prime containing exactly 2 more digits on each end than a(n-1), with a(n-1) as the central substring.

Original entry on oeis.org

3, 10301, 101030101, 1210103010121, 12121010301012121, 111212101030101212111, 3111121210103010121211113, 17311112121010301012121111371, 961731111212101030101212111137169, 3196173111121210103010121211113716913, 95319617311112121010301012121111371691359, 109531961731111212101030101212111137169135901, 1410953196173111121210103010121211113716913590141, 13141095319617311112121010301012121111371691359014131

Offset: 1

Shyam Sunder Gupta, Aug 24 2024

This is a finite sequence since at a(14) there is no way to add 2 more digits and reach a palindromic prime.

			As a triangle:
          3
        10301
      101030101
    1210103010121
  12121010301012121

Cf. A047076, A053600, A082563.

from sympy import isprime
from itertools import product
def agen(): # generator of terms
    an, s = 3, "3"
    while an > 0:
        yield an
        an = -1
        for f, r in product("1379", "0123456789"):
            sn = f+r+s+r+f
            if isprime(t:=int(sn)):
                an, s = t, sn
                break
print(list(agen())) # Michael S. Branicky, Aug 25 2024

cp's OEIS Frontend

A052092 Lengths of the palindromic primes from Honaker's sequence A053600.

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A261881 Minimal nested palindromic primes with seed 0.

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A047076 a(n+1) is the smallest palindromic prime containing exactly 2 more digits on each end than the previous term, with a(n) as a central substring.

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A052205 a(n+1) is smallest palindromic prime containing exactly 3 more digits on each end than the previous term, with a(n) as a central substring.

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A052091 Left parts needed for the construction of the palindromic prime pyramid starting with 2.

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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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A256957 Smallest palindromic prime that generates a palindromic prime pyramid of height n.

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A034276 Smallest prime that generates a prime pyramid of height n.

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A247483 a(1)=4; for n>=1, a(n+1) is the smallest palindromic semiprime with a(n) as a central substring.

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