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a(1) = 2; for n>=1, a(n+1) is the smallest palindromic prime with a(n) as a central substring.
13

%I #33 Aug 11 2024 14:41:32

%S 2,727,37273,333727333,93337273339,309333727333903,

%T 1830933372733390381,92183093337273339038129,

%U 3921830933372733390381293,1333921830933372733390381293331,18133392183093337273339038129333181

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

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

%H Clark Kimberling, <a href="/A053600/b053600.txt">Table of n, a(n) for n = 1..200</a>

%H P. De Geest, <a href="https://www.worldofnumbers.com/palprim3.htm">Palindromic Prime Pyramid Puzzle by G.L.Honaker,Jr</a>

%H G. L. Honaker, Jr. and Chris Caldwell, <a href="https://t5k.org/curios/cpage/1143.html">Prime Curios! 18133...33181 (35-digits)</a>

%H G. L. Honaker, Jr. & C. K. Caldwell, <a href="http://www.utm.edu/staff/caldwell/preprints/JRM_prime_pyramids.pdf">Palindromic Prime Pyramids</a>

%H G. L. Honaker, Jr. & C. K. Caldwell, <a href="http://www.utm.edu/staff/caldwell/supplements">Supplement to "Palindromic Prime Pyramids"</a>

%H Ivars Peterson, <a href="https://www.sciencenews.org/article/primes-palindromes-and-pyramids">Primes, Palindromes, and Pyramids</a>, Science News.

%H Inder J. Taneja, <a href="https://www.researchgate.net/profile/Inder_Taneja/publication/319151146_Palindromic_Prime_Embedded_Trees/links/5994fddb458515c0ce653c73/Palindromic-Prime-Embedded-Trees.pdf">Palindromic Prime Embedded Trees</a>, RGMIA Res. Rep. Coll. 20 (2017), Art. 124.

%H Inder J. Taneja, <a href="https://rgmia.org/papers/v21/v21a75.pdf">Same Digits Embedded Palprimes</a>, RGMIA Research Report Collection (2018) Vol. 21, Article 75, 1-47.

%e As a triangle:

%e .........2

%e ........727

%e .......37273

%e .....333727333

%e ....93337273339

%e ..309333727333903

%e 1830933372733390381

%t 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 *)

%o (Python)

%o from gmpy2 import digits, mpz, is_prime

%o A053600_list, p = [2], 2

%o for _ in range(30):

%o ....m, ps = 1, digits(p)

%o ....s = mpz('1'+ps+'1')

%o ....while not is_prime(s):

%o ........m += 1

%o ........ms = digits(m)

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

%o ....p = s

%o ....A053600_list.append(int(p)) # _Chai Wah Wu_, Apr 09 2015

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

%K base,nonn

%O 1,1

%A _G. L. Honaker, Jr._, Jan 20 2000