A087166 Primes which are palindromes in 3 or more bases.

17, 31, 67, 73, 107, 109, 127, 151, 157, 173, 181, 191, 197, 211, 227, 241, 257, 271, 277, 307, 313, 337, 353, 373, 379, 401, 409, 419, 421, 433, 443, 457, 461, 463, 487, 521, 523, 541, 577, 587, 601, 617, 619, 631, 647, 661, 673, 683, 701, 719, 727, 743, 757, 761, 773, 787, 797, 809, 857, 859

Offset: 1

Author

Randy L. Ekl, Oct 18 2003

Keywords

• base
• nonn

Comments

For the purposes of this sequence, single digits are not counted as palindromes (otherwise every number n is a palindrome in all bases > n). - Robert Israel, May 01 2020

Examples

			31 is in the list, as 31 base 2 = 11111, 31 base 5 = 111 and 31 base 30 = 11, i.e. three different ways.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..5867

Crossrefs

Primes in A253594.

Programs

Maple

N:= 1000: # for all terms <= N
digrev:= proc(n,b)
  local L,i;
  L:= convert(n,base,b);
  add(L[-i]*b^(i-1),i=1..nops(L))
end proc:
bpalis:= proc(b, N)
  local Res,dmax,d,m;
  dmax:= floor(log[b](N))+1;
  if dmax < 2 then return [] fi;
  Res:= seq(i*(b+1),i=1..b-1);
  for d from 3 to dmax do
    if d::even then
      m:= d/2;
      Res:= Res, seq(n*b^m + digrev(n,b),n=b^(m-1)..b^m-1);
    else
      m:= (d-1)/2;
      Res:= Res, seq(seq(n*b^(m+1)+y*b^m+digrev(n,b), y=0..b-1), n=b^(m-1)..b^m-1);
    fi
  od;
  select(`<=`,[Res], N)
end proc:
V:= Vector(N):
for b from 2 to N-1 do
  bp:= bpalis(b,N);
  V[bp]:= V[bp] +~ 1
od:
select(p -> isprime(p) and V[p] >= 3, [seq(i,i=3..N,2)]); # Robert Israel, May 01 2020

Extensions

Corrected by Robert Israel, May 01 2020