A034276 -id:A034276 - cp's OEIS frontend

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

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

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

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

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

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