1,3

All elements of the sequence greater than 6 are prime (ab=a(b-1)+a or a^2=(a-1)^2+2(a-1)+1). Mersenne and Fermat primes are not in the sequence.

Additional comments: if you can factor a number as a*b then it is a palindrome in base b-1, where b is the larger of the two factors. (If the number is a square, then it can be a palindrome in an additional way, in base (sqrt(n)-1)). The ab form does not work when a = b-1, but of course there are no two consecutive primes (other than 2,3, which explains the early special cases), so if you can factor a number as a(a-1), then another factorization also exists). - Michael B Greenwald (mbgreen(AT)central.cis.upenn.edu), Jan 01, 2002

Note that no prime p is palindromic in base b for the range sqrt(p) < b < p-1. Hence to find nonpalindromic primes, we need only examine bases up to floor(sqrt(p)), which greatly reduces the computational effort required. - T. D. Noe, Mar 01 2008

Paul Guinand, Strictly non-palindromic numbers, unpublished note, 1996.

T. D. Noe, Table of n, a(n) for n=1..10001

K. S. Brown, On General Palindromic Numbers

P. De Geest, Palindromic numbers beyond base 10

John P. Linderman, Description of A135549-A016038

John P. Linderman, Perl program [Use the command: HASNOPALINS=1 palin.pl]

PalindromicQ[n_, base_] := FromDigits[Reverse[IntegerDigits[n, base]], base] == n; PalindromicBases[n_] := Select[Range[2, n-2], PalindromicQ[n, # ] &]; StrictlyPalindromicQ[n_] := PalindromicBases[n] == {}; Select[Range[150], StrictlyPalindromicQ] - Herman Beeksma (herman(AT)beeksma.nl), Jul 16 2005

palindromicBases[n_] := Module[{p}, Table[p = IntegerDigits[n, b]; If[ p == Reverse[p], {b, p}, Sequence @@ {}], {b, 2, n - 2}]]; lst = {0, 1, 4, 6}; Do[ If[ Length@ palindromicBases@ Prime@n == 0, AppendTo[lst, Prime@n]], {n, 10000}]; lst (from Robert G. Wilson v (rgwv(AT)rgwv.com), Mar 08 2008)

Cf. A047811, A050813, A050812, A135550, A135551, A135549.

Cf. A037183, A135550, A135551, A135549, A138348.

Sequence in context: A111124 A117308 A114412 this_sequence A003099 A061941 A029505

Adjacent sequences: A016035 A016036 A016037 this_sequence A016039 A016040 A016041

nonn,base,nice

Robert G. Wilson v (rgwv(AT)rgwv.com)

Extended and corrected by Patrick De Geest (pdg(AT)worldofnumbers.com), Oct 15 1999.

Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 09 2008