A258372 - OEIS

%I #33 Nov 09 2019 08:19:29

%S 0,3,4,8,36,8,5,72,28,6,79,212,23,6,73,24,52,62,3,28,220,53,75,58,228,

%T 9,265,89,214,86,215,4,7,39,295,40,87,216,97,6,264,53,287,223,4,239,

%U 259,25,57,364,49,38,93,86,27,30,80,24,6,356,50,645,395,206

%N Smallest nonnegative number k not starting or ending with the digit 1 that forms a prime when it is sandwiched between n ones to the left of k and n ones to the right of k.

%C n = 1 is the only case where a(n) = 0, since for any n > 1, A138148(n) is divisible by A002275(n).

%C No n exists such that a(n) = 2, since any number of the form A100706(n)+A011557(n) is of the form A000533(n)*A002275(n+1) (see comment by Robert Israel in A107123).

%C a(n) = 3 iff n is in A107123.

%C a(n) = 4 iff n is in A107124.

%C If k has an even number of digits and is a multiple of 11, then k is not a term. If k = (10^r+1)(10^m-1)/9 for some m > 0, r >= 0, then k is not a term. If A272232(k) = 0, then k is not a term. - _Chai Wah Wu_, Nov 08 2019

%H Giovanni Resta, <a href="/A258372/b258372.txt">Table of n, a(n) for n = 1..1000</a>

%e a(1) = 0, because 101 is prime.

%e a(5) = 36, because the smallest x >= 0 such that 11111_x_11111 (where '_' denotes concatenation) is prime is 36. The decimal expansion of that prime is 111113611111.

%t Table[k = 0; s = Table[1, {n}]; While[Or[!PrimeQ[FromDigits[s ~Join~ IntegerDigits[k] ~Join~ s]], Or[First@ IntegerDigits@ k == 1, Last@ IntegerDigits@ k == 1]], k++]; k, {n, 64}] (* _Michael De Vlieger_, May 28 2015 *)

%o (PARI) a000042(n) = (10^n-1)/9

%o a(n) = my(k=0); while(k==10 || k%10==1 || k\(10^(#Str(k)-1))==1 || !ispseudoprime(eval(Str(a000042(n), k, a000042(n)))), k++); k

%Y Cf. A088281, A090287, A272232.

%K nonn,base

%O 1,2

%A _Felix Fröhlich_, May 28 2015