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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.

%I A053600 #37 Nov 06 2024 15:26:46
%S A053600 2,727,37273,333727333,93337273339,309333727333903,
%T A053600 1830933372733390381,92183093337273339038129,
%U A053600 3921830933372733390381293,1333921830933372733390381293331,18133392183093337273339038129333181
%N A053600 a(1) = 2; for n>=1, a(n+1) is the smallest palindromic prime with a(n) as a central substring.
%D A053600 G. L. Honaker, Jr. and Chris K. Caldwell, Palindromic Prime Pyramids, J. Recreational Mathematics, Vol. 30(3) 169-176, 1999-2000.
%H A053600 Clark Kimberling, <a href="/A053600/b053600.txt">Table of n, a(n) for n = 1..200</a>
%H A053600 P. De Geest, <a href="https://www.worldofnumbers.com/palprim3.htm">Palindromic Prime Pyramid Puzzle by G.L.Honaker,Jr</a>
%H A053600 G. L. Honaker, Jr. and Chris Caldwell, <a href="https://t5k.org/curios/cpage/1143.html">Prime Curios! 18133...33181 (35-digits)</a>
%H A053600 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 A053600 G. L. Honaker, Jr. & C. K. Caldwell, <a href="http://www.utm.edu/staff/caldwell/supplements">Supplement to "Palindromic Prime Pyramids"</a>
%H A053600 Ivars Peterson, <a href="https://www.sciencenews.org/article/primes-palindromes-and-pyramids">Primes, Palindromes, and Pyramids</a>, Science News.
%H A053600 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 A053600 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 A053600 As a triangle:
%e A053600 .........2
%e A053600 ........727
%e A053600 .......37273
%e A053600 .....333727333
%e A053600 ....93337273339
%e A053600 ..309333727333903
%e A053600 1830933372733390381
%t A053600 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 A053600 (Python)
%o A053600 from gmpy2 import digits, mpz, is_prime
%o A053600 A053600_list, p = [2], 2
%o A053600 for _ in range(30):
%o A053600     m, ps = 1, digits(p)
%o A053600     s = mpz('1'+ps+'1')
%o A053600     while not is_prime(s):
%o A053600         m += 1
%o A053600         ms = digits(m)
%o A053600         s = mpz(ms+ps+ms[::-1])
%o A053600     p = s
%o A053600     A053600_list.append(int(p)) # _Chai Wah Wu_, Apr 09 2015
%Y A053600 Cf. A000040, A002385, A047076, A052205, A034276, A256957, A052091, A052092, A261881.
%K A053600 base,nonn
%O A053600 1,1
%A A053600 _G. L. Honaker, Jr._, Jan 20 2000