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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 a*b 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 non-palindromic primes, we need only examine bases up to floor(sqrt(p)), which greatly reduces the computational effort required. - T. D. Noe, Mar 01 2008

No number n is palindromic in any base b with n/2 <= b <= n-2, so this is also numbers not palindromic in any base b with 2 <= b <= n/2.

Sequence A047811 (this sequence without 0, 1, 2, 3) is mentioned in the Guy paper, in which he reports on unsolved problems. This problem came from Mario Borelli and Cecil B. Mast. The paper poses two questions about these numbers: (1) Can palindromic or nonpalindromic primes be otherwise characterized? and (2) What is the cardinality, or the density, of the set of palindromic primes? Of the set of nonpalindromic primes? - T. D. Noe, Apr 18 2011

From Robert G. Wilson v, Oct 22 2014 and Nov 03 2014: (Start)

Define f(n) to be the number of palindromic representations of n in bases b with 1 < b < n, see A135551.

For A016038, f(n) = 1 for all n. Only the numbers n = 0, 1, 4 and 6 are not primes.

For f(n) = 2, all terms are prime or semiprimes (prime omega <= 2 (A037143)) with the exception of 8 and 12;

For f(n) = 3, all terms are at most 3-almost primes (prime omega <= 3 (A037144)), with the exception of 16, 32, 81 and 625;

For f(n) = 4, all terms are at most 4-almost primes, with the exception of 64 and 243;

For f(n) = 5, all terms are at most 5-almost primes, with the exception of 128, 256 and 729;

For f(n) = 6, all terms are at most 6-almost primes, with the sole exception of 2187;

For f(n) = 7, all terms are at most 7-almost primes, with the exception of 512, 2048 and 19683; etc. (End)

a(1) = 1 by convention, which makes this sequence different from A135551.

Index of first occurrence of k in A037183. - Robert G. Wilson v, Oct 27 2014

a(n) >= 1, since n will always have a single "digit" in base n+1.

The strictly non-palindromic twin primes are a part of the normal twin primes. See the list of twin primes A077800 and A016038 for the strictly non-palindromic numbers.

Up to 10^9 there are 2992 triples of strictly non-palindromic primes if the quadruples and quintuples are not counted.

For quadruples of this kind, see A138359.

For quintuples of this kind, see A138360.

For triples of this kind, see A138358.

For quintuples of this kind, see A138360.

The quintuples T(n,1), T(n,2), .. T(n,5), n>=1, in this array are 5 consecutive primes (consecutive in A000040) which are also members of A016038.

Notice that all strictly non-palindromic numbers >6 are prime! (See A016038.) Quintuples of these strictly non-palindromic primes, without any normal prime in between, are listed here.

Up to 1 billion there are only 5 quintuples of strictly non-palindromic primes. May be that there are no more quintuples of this kind. Up to 1 billion there are no n-tuples of strictly non-palindromic primes with n>5.