A087155 Primes having nontrivial palindromic representation in some (at least one) base.

5, 7, 13, 17, 23, 29, 31, 37, 41, 43, 59, 61, 67, 71, 73, 83, 89, 97, 101, 107, 109, 113, 127, 131, 151, 157, 173, 181, 191, 193, 197, 199, 211, 227, 229, 233, 239, 241, 251, 257, 271, 277, 281, 307, 313, 331, 337, 349, 353, 373, 379, 383, 397, 401, 409, 419

Offset

1,1

Comments

Number of terms < 10^n: 2, 18, 129, 1010, 8392, ..., . - Robert G. Wilson v, Jun 19 2014

Every whole number has single-digit representation in all large bases and all greater than 2 have representation 11 in the base one less than itself. Other palindromic representations are the nontrivial ones. - James G. Merickel, Jul 25 2015

Primes not in A016038. - Robert Israel, Jul 27 2015

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Example

17 is in the list because 17_2 = 10001 and 17_4 = 101, two nontrivial palindromic representations. 19 is not in the list because 19 is not a multidigit palindrome in any base other than base 18.

Maple

filter:= proc(n) local b, L;
if not isprime(n) then return false fi;
for b from 2 to floor(sqrt(n)) do
  L:= convert(n, base, b);
  if L = ListTools:-Reverse(L) then return true fi;
od:
false
end proc:
select(filter, [2*i+1 $ i=1..1000]); # Robert Israel, Jul 27 2015

Mathematica

palindromicBases[n_] := Module[{p}, Table[p = IntegerDigits[n, b]; If[p == Reverse[p], {b, p}, Sequence @@ {}], {b, 2, n - 2}]]; Select[ Prime@ Range@ 300, palindromicBases[#] !={}&] (* Robert G. Wilson v, May 06 2014 *)

Prog

(PARI) q=1; forprime(m=3, 500, count=0; for(b=2, m-1, w=b+1; k=0; i=m; while(i>0, k=k*w+i%b; i=floor(i/b)); l=0; j=k; while(j>0, l=l*w+j%w; j=floor(j/w)); if(l==k, count=count+1; if(count>1, print1(m, ", "); q=b; m=nextprime(m+1); q=1; b=1, q=b), )))

Crossrefs

Cf. A016038.

Sequence in context: A240716 A216773 A216745 * A216753 A045442 A216777

Adjacent sequences: A087152 A087153 A087154 * A087156 A087157 A087158

Keyword

base,nonn

Author

Randy L. Ekl, Oct 18 2003

Extensions

Title, comments and example changed to agree with convention on single-digit numbers and incorporate 'nontrivial' concept by James G. Merickel, Jul 25 2015

Status

approved