A047811 Numbers n >= 4 that are not palindromic in any base b, 2 <= b <= n/2.

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

N. J. A. Sloane

Keywords

• nonn
• base
• easy
• nice

Comments

Sequence A016038 is identical up to four additional terms: 0, 1, 2, 3; see there for more information.

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

This sequence is mentioned in the paper by Richard Guy, 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 17 2011

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• R. K. Guy, Conway's RATS and other reversals, Amer. Math. Monthly, 96 (1989), 425-428.

Crossrefs

Cf. A016038, A050812, A050813.

Cf. A135549.

Programs

Mathematica

Select[Range[4,1500],And@@(#!=Reverse[#]&/@Table[IntegerDigits[#,b],{b,2,#/2}])&] (* Harvey P. Dale, May 22 2013 *)

PARI

is(n)=!for(b=2,n\2,Vecrev(d=digits(n,b))==d&&return)&&n>3 \\ M. F. Hasler, Sep 08 2015

Formula

a(n) = A016038(n+4) for all n. - M. F. Hasler, Sep 08 2015

Extensions

Extended (and corrected) by Patrick De Geest, Oct 15 1999

Minor edits by M. F. Hasler, Sep 08 2015