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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

REFERENCES

Paul Guinand, Strictly non-palindromic numbers, unpublished note, 1996.

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

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

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 (* from 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 July 25 21:57 EDT 2014. Contains 244919 sequences.