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A090287 Smallest prime obtained by sandwiching n between a number with identical digits, or 0 if no such prime exists. Primes of the form k n k where all the digits of k are identical.

%I A090287 #38 Apr 10 2025 23:23:06
%S A090287 101,313,727,131,11411,151,777767777,373,181,191,9109,0,7127,331333,
%T A090287 991499,1151,3163,1171,1181,9199,1201,112111,0,1231,7247,3253,
%U A090287 7777777777267777777777,1111271111,11128111,1291,1301,3313,1321,0,3343,333533,1361,3373,1381
%N A090287 Smallest prime obtained by sandwiching n between a number with identical digits, or 0 if no such prime exists. Primes of the form k n k where all the digits of k are identical.
%C A090287 a(n) = 0 if n is a palindrome with even number of digits. Conjecture: No other term is zero.
%C A090287 The conjecture is false.  a(231) = 0, a(420) = 0, a(n) = 0 if 11 divides n and n has an even number of digits. a(1414) has over 2000 digits. - _Chai Wah Wu_, Mar 31 2015
%H A090287 Chai Wah Wu, <a href="/A090287/b090287.txt">Table of n, a(n) for n = 0..365</a>
%H A090287 Chai Wah Wu, <a href="http://arxiv.org/abs/1503.08883">On a conjecture regarding primality of numbers constructed from prepending and appending identical digits</a>, arXiv:1503.08883 [math.NT], 2015.
%H A090287 <a href="/index/Pri#piden">Index entries for primes involving decimal expansion of n</a>
%t A090287 (* f(n) defined by _José de Jesús Camacho Medina_ in A010785. *)
%t A090287 lst={};f[m_]:=IntegerDigits[(m-9*Floor[(m-1)/9])*(10^Floor[(m+8)/9]-1)/9];
%t A090287 g[n_]:=FromDigits[Flatten[{f[m],IntegerDigits[n],f[m]}]];
%t A090287 Do[m=1;While[True,If[Mod[Length[IntegerDigits[n]],2]==0&&IntegerDigits[n]==Reverse[IntegerDigits[n]],
%t A090287 AppendTo[lst,0];Break[],If[PrimeQ[g[n]],AppendTo[lst,g[n]];Break[]]];m++],{n,25}];
%t A090287 lst (* _Ivan N. Ianakiev_, Mar 23 2015 *)
%o A090287 (Python)
%o A090287 from gmpy2 import is_prime, mpz, digits
%o A090287 def A090287(n, limit=2000):
%o A090287     sn = str(n)
%o A090287     if n in (231, 420, 759) or not (len(sn) % 2 or n % 11):
%o A090287         return 0
%o A090287     for i in range(1, limit+1):
%o A090287         for j in range(1, 10, 2):
%o A090287             si = digits(j, 10)*i
%o A090287             p = mpz(si+sn+si)
%o A090287             if is_prime(p):
%o A090287                 return int(p)
%o A090287     else:
%o A090287         return 'search limit reached.' # _Chai Wah Wu_, Mar 31 2015
%Y A090287 Cf. A010785, A338712.
%K A090287 base,nonn
%O A090287 0,1
%A A090287 _Amarnath Murthy_, Nov 29 2003
%E A090287 a(0) from _Chai Wah Wu_, Mar 23 2015
%E A090287 a(26)-a(38) from _Chai Wah Wu_, Mar 24 2015