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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.
2
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
1,1
COMMENTS
All positive integers that are palindromes when written in binary are exclusively either in this sequence or in sequence A163410.
LINKS
EXAMPLE
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.
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 -> ormap(not isbpali, numtheory:-factorset(t)), Bp):
sort(convert(R, list)); # Robert Israel, Dec 19 2016
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
base,nonn
AUTHOR
Leroy Quet, Jul 27 2009
EXTENSIONS
More terms from Stefan Steinerberger, Aug 05 2009
Corrected by Leroy Quet, Aug 09 2009
STATUS
approved