A016038 Strictly non-palindromic numbers: n is not palindromic in any base b with 2 <= b <= n-2.

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

Offset: 1

Author

Robert G. Wilson v

Keywords

• nonn
• base
• nice
• easy

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 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)

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
• Patrick 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]

Crossrefs

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

Programs

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 *)
Select[Range@ 1500, Function[n, NoneTrue[Range[2, n - 2], PalindromeQ@ IntegerDigits[n, #] &]]] (* Michael De Vlieger, Dec 24 2017 *)

PARI

is(n)=!for(b=2,n\2,Vecrev(d=digits(n,b))==d&&return) \\ M. F. Hasler, Sep 08 2015

Python

from itertools import count, islice
from sympy.ntheory.factor_ import digits
def A016038_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n: all((s := digits(n,b)[1:])[:(t:=len(s)+1>>1)]!=s[:-t-1:-1] for b in range(2,n-1)), count(max(startvalue,0)))
A016038_list = list(islice(A016038_gen(),30)) # Chai Wah Wu, Jan 17 2024

Formula

a(n) = A047811(n-4) for n > 4. - M. F. Hasler, Sep 08 2015

Extensions

Extended and corrected by Patrick De Geest, Oct 15 1999

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