A331807 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.

3, 13, 5, 31, 7, 71, 73, 109, 11, 199, 157, 313, 197, 241, 17, 307, 19, 419, 401, 463, 23, 599, 577, 701, 677, 757, 29, 929, 991, 1117, 1153, 1123, 1259, 1471, 37, 1481, 1483, 1873, 41, 1723, 43, 1979, 2069, 2161, 47, 2351, 2593, 2549, 2551, 2857, 53, 2969, 2917, 3191, 3137

Offset

2,1

Comments

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):

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.

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

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).

Links

Chai Wah Wu, Table of n, a(n) for n = 2..10000

Example

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.
If we use the representation described earlier, then:
   a(2)  = 3    is [1, 1]_2,
   a(3)  = 13   is [1, 1, 1]_3,
   a(4)  = 5    is [1, 1]_4,
   a(11) = 199  is [1, 7, 1]_11,
   a(13) = 313  is [1, 11, 1]_13,
   a(16) = 17   is [1, 1]_16,
   a(48) = 2593 is [1, 6, 1]_48.

Mathematica

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

Crossrefs

A331806 is a similar sequence where repeated terms are allowed.

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

Sequence in context: A125571 A187023 A331806 * A084738 A352256 A073580

Adjacent sequences: A331804 A331805 A331806 * A331808 A331809 A331810

Keyword

nonn,base,easy

Author

Colin M Ready, Feb 22 2020

Status

approved