A207039 -id:A207039 - cp's OEIS frontend

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

Leroy Quet, Jul 27 2009

			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.

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

Cf. A006995, A163411, A207039.

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

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

More terms from Sean A. Irvine, Nov 10 2009

A163411 A positive integer is included if it is a palindrome when written in binary, and it is divisible by at least one prime that is not a binary palindrome.

Original entry on oeis.org

33, 65, 99, 129, 165, 195, 231, 273, 297, 325, 341, 387, 403, 427, 455, 471, 495, 513, 561, 585, 633, 645, 693, 717, 819, 843, 891, 903, 951, 975, 1023, 1025, 1057, 1105, 1137, 1161, 1273, 1317, 1365, 1397, 1421, 1501, 1539, 1651, 1675, 1707

Offset: 1

Leroy Quet, Jul 27 2009

All positive integers that are palindromes when written in binary are exclusively either in this sequence or in sequence A163410.

			99 in binary is 1100011, which is a palindrome. 99 is divisible by the primes 3 and 11. 3 in binary is 11, a palindrome. But 11(decimal) in binary is 1011, not a palindrome. Since there is at least one prime dividing the binary palindrome 99 that is not a binary palindrome, then 99 is included in this sequence.

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

Cf. A006995, A163410, A207039.

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 -> ormap(not isbpali, numtheory:-factorset(t)), Bp):
sort(convert(R,list)); # Robert Israel, Dec 19 2016

a = {}; For[n = 2, n < 10000, n++, If[FromDigits[Reverse[IntegerDigits[n, 2]], 2] == n, b = Table[FactorInteger[n][[i, 1]], {i, 1, Length[FactorInteger[n]]}]; For[i = 1, i < Length[b] + 1, i++, If[ ! FromDigits[Reverse[IntegerDigits[b[[i]], 2]], 2] == b[[i]], AppendTo[a, n]; Break]]]]; a (* Stefan Steinerberger, Aug 05 2009 *)

More terms from Stefan Steinerberger, Aug 05 2009

Corrected by Leroy Quet, Aug 09 2009

cp's OEIS Frontend

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.

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