A331807 - OEIS

%I #15 Jul 03 2020 02:37:57

%S 3,13,5,31,7,71,73,109,11,199,157,313,197,241,17,307,19,419,401,463,

%T 23,599,577,701,677,757,29,929,991,1117,1153,1123,1259,1471,37,1481,

%U 1483,1873,41,1723,43,1979,2069,2161,47,2351,2593,2549,2551,2857,53,2969,2917,3191,3137

%N a(n) is the smallest prime number p > n, not yet in the sequence, such that p is a palindrome when written in base n.

%C Using a representation where the digits of the prime are written between "[" and "]_" separated by commas with the base following the "_" then by checking up to a base of 7000 (where the lowest prime palindrome is [1, 1]_7000):

%C 1) Either the palindrome is [1, 1]_n where n is one less than a prime number, or [1, X, 1]_n where X << n, asymptotically.

%C 2) A conjecture: No lowest primes need more than three digits.

%C 3) The terms a(12) and a(30) differ from the similar sequence A331806 as these terms in A331806 are the same as the earlier terms a(3) and a(5).

%H Chai Wah Wu, <a href="/A331807/b331807.txt">Table of n, a(n) for n = 2..10000</a>

%e a(2)=3 which is 11 in binary, a(3)=13 which is 111 in ternary, a(4)=5 which is 11 in quaternary, a(16)=17 which is 11 in hexadecimal.

%e If we use the representation described earlier, then:

%e    a(2)  = 3    is [1, 1]_2,

%e    a(3)  = 13   is [1, 1, 1]_3,

%e    a(4)  = 5    is [1, 1]_4,

%e    a(11) = 199  is [1, 7, 1]_11,

%e    a(13) = 313  is [1, 11, 1]_13,

%e    a(16) = 17   is [1, 1]_16,

%e    a(48) = 2593 is [1, 6, 1]_48.

%t Array[Block[{p = Prime[PrimePi[#] + 1]}, While[! PalindromeQ@ IntegerDigits[p, #], p = NextPrime@ p]; p] &, 55, 2] (* _Michael De Vlieger_, Feb 25 2020 *)

%Y A331806 is a similar sequence where repeated terms are allowed.

%Y Cf. A006093 (prime(n) - 1).

%K nonn,base,easy

%O 2,1

%A _Colin M Ready_, Feb 22 2020