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A016038 Strictly non-palindromic numbers: n is not palindromic in any base b with 2 <= b <= n-2. 17
0, 1, 2, 3, 4, 6, 11, 19, 47, 53, 79, 103, 137, 139, 149, 163, 167, 179, 223, 263, 269, 283, 293, 311, 317, 347, 359, 367, 389, 439, 491, 563, 569, 593, 607, 659, 739, 827, 853, 877, 977, 983, 997, 1019, 1049, 1061, 1187, 1213, 1237, 1367, 1433, 1439, 1447, 1459 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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 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)

REFERENCES

Paul Guinand, Strictly non-palindromic numbers, unpublished note, 1996.

LINKS

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

R. K. Guy, Conway's RATS and other reversals, Amer. Math. Monthly, 96 (1989), 425-428.

John P. Linderman, Description of A135549-A016038

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

MATHEMATICA

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, 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 (* Robert G. Wilson v, Mar 08 2008 *)

CROSSREFS

Cf. A047811, A050813, A050812, A050813, A135550, A135551, A135549, A037183, A135550, A135551, A135549, A138348, A033868.

Sequence in context: A111124 A117308 A114412 * A003099 A061941 A029505

Adjacent sequences:  A016035 A016036 A016037 * A016039 A016040 A016041

KEYWORD

nonn,base,nice,easy

AUTHOR

Robert G. Wilson v

EXTENSIONS

Extended and corrected by Patrick De Geest, Oct 15 1999

Edited by N. J. A. Sloane, Apr 09 2008

STATUS

approved

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Last modified November 22 05:19 EST 2014. Contains 249804 sequences.