|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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):
|
|
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
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|