A163410 A positive integer is included if it is a palindrome when written in binary, and it is not divisible by any primes that are not binary palindromes.

1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 45, 51, 63, 73, 85, 93, 107, 119, 127, 153, 189, 219, 255, 257, 313, 365, 381, 443, 511, 765, 771, 1193, 1241, 1285, 1453, 1533, 1571, 1619, 1787, 1799, 1831, 1879, 2313, 3579, 3855, 4369, 4889, 5113, 5189, 5397, 5557, 5869

Offset: 1

Author

Leroy Quet, Jul 27 2009

Keywords

• base
• nonn

Examples

			51 in binary is 110011, which is a palindrome. 51 is divisible by the primes 3 and 17. 3 in binary is 11, a palindrome. And 17 in binary is 10001, also a palindrome. Since all the primes dividing the binary palindrome 51 are themselves binary palindromes, then 51 is included in this sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006995, A163411, A207039.

Programs

Maple

dmax:= 15: # to get all terms with at most dmax binary digits
revdigs:= proc(n)
  local L, Ln, i;
  L:= convert(n, base, 2);
  Ln:= nops(L);
  add(L[i]*2^(Ln-i), i=1..Ln);
end proc:
isbpali:= proc(n) option remember; local L; L:= convert(n,base,2); L=ListTools:-Reverse(L) end proc:
Bp:= {0, 1}:
for d from 2 to dmax do
  if d::even then
    Bp:= Bp union {seq(2^(d/2)*x + revdigs(x), x=2^(d/2-1)..2^(d/2)-1)}
  else
    m:= (d-1)/2;
    B:={seq(2^(m+1)*x + revdigs(x), x=2^(m-1)..2^m-1)};
    Bp:= Bp union B union map(`+`, B, 2^m)
  fi
od:
R:= select(t -> andmap(isbpali, numtheory:-factorset(t)), Bp minus {0}):
sort(convert(R,list)); # Robert Israel, Dec 19 2016

Mathematica

binPalQ[n_] := PalindromeQ @ IntegerDigits[n, 2]; Select[Range[6000], binPalQ[#] && AllTrue[FactorInteger[#][[;; , 1]], binPalQ] &] (* Amiram Eldar, Mar 30 2021 *)

Extensions

More terms from Sean A. Irvine, Nov 10 2009