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A035928
Numbers n such that BCR(n) = n, where BCR = binary-complement-and-reverse = take one's complement then reverse bit order.
34
2, 10, 12, 38, 42, 52, 56, 142, 150, 170, 178, 204, 212, 232, 240, 542, 558, 598, 614, 666, 682, 722, 738, 796, 812, 852, 868, 920, 936, 976, 992, 2110, 2142, 2222, 2254, 2358, 2390, 2470, 2502, 2618, 2650, 2730, 2762, 2866, 2898, 2978, 3010, 3132, 3164, 3244
OFFSET
1,1
COMMENTS
Numbers n such that A036044(n) = n.
Also: numbers such that n+BR(n) is in A000225={2^k-1} (with BR = binary reversed). - M. F. Hasler, Dec 17 2007
Also called "antipalindromes". - Jeffrey Shallit, Feb 04 2022
LINKS
James Haoyu Bai, Joseph Meleshko, Samin Riasat, and Jeffrey Shallit, Quotients of Palindromic and Antipalindromic Numbers, arXiv:2202.13694 [math.NT], 2022.
Aayush Rajasekaran, Jeffrey Shallit, and Tim Smith, Sums of Palindromes: an Approach via Nested-Word Automata, preprint arXiv:1706.10206 [cs.FL], June 30 2017.
FORMULA
If offset were 0, a(2n+1) - a(2n) = 2^floor(log_2(n)+1).
a(n) = n * A062383(n) + A036044(n). - Rémy Sigrist, Jun 11 2022
EXAMPLE
38 is such a number because 38=100110; complement to get 011001, then reverse bit order to get 100110.
MAPLE
[seq(ReflectBinSeq(j, (floor_log_2(j)+1)), j=1..256)];
ReflectBinSeq := (x, n) -> (((2^n)*x)+binrevcompl(x));
binrevcompl := proc(nn) local n, z; n := nn; z := 0; while(n <> 0) do z := 2*z + ((n+1) mod 2); n := floor(n/2); od; RETURN(z); end;
floor_log_2 := proc(n) local nn, i: nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi: nn := floor(nn/2); od: end; # Computes essentially the same as floor(log[2](n))
# alternative Maple program:
q:= n-> (l-> is(n=add((1-l[-i])*2^(i-1), i=1..nops(l))))(Bits[Split](n)):
select(q, [$1..3333])[]; # Alois P. Heinz, Feb 10 2021
MATHEMATICA
bcrQ[n_]:=Module[{idn2=IntegerDigits[n, 2]}, Reverse[idn2/.{1->0, 0->1}] == idn2]; Select[Range[3200], bcrQ] (* Harvey P. Dale, May 24 2012 *)
PROG
(PARI) for(n=1, 1000, l=length(binary(n)); b=binary(n); if(sum(i=1, l, abs(component(b, i)-component(b, l+1-i)))==l, print1(n, ", ")))
(PARI) for(i=1, 999, if(Set(vecextract(t=binary(i), "-1..1")+t)==[1], print1(i", "))) \\ M. F. Hasler, Dec 17 2007
(PARI) a(n) = my (b=binary(n)); (n+1)*2^#b-fromdigits(Vecrev(b), 2)-1 \\ Rémy Sigrist, Mar 15 2021
(Haskell)
a035928 n = a035928_list !! (n-1)
a035928_list = filter (\x -> a036044 x == x) [0, 2..]
-- Reinhard Zumkeller, Sep 16 2011
(Python)
def comp(s): z, o = ord('0'), ord('1'); return s.translate({z:o, o:z})
def BCR(n): return int(comp(bin(n)[2:])[::-1], 2)
def aupto(limit): return [m for m in range(limit+1) if BCR(m) == m]
print(aupto(3244)) # Michael S. Branicky, Feb 10 2021
(Python)
from itertools import count, islice
def A035928_gen(startvalue=1): # generator of terms >= startvalue
return filter(lambda n:n==int(format(~n&(1<<(m:=n.bit_length()))-1, '0'+str(m)+'b')[::-1], 2), count(max(startvalue, 1)))
A035928_list = list(islice(A035928_gen(), 30)) # Chai Wah Wu, Jun 30 2022
CROSSREFS
Cf. A061855.
Intersection of A195064 and A195066; cf. A195063, A195065.
Sequence in context: A176978 A186630 A154391 * A014486 A166751 A216649
KEYWORD
nonn,nice,easy,base
AUTHOR
EXTENSIONS
More terms from Erich Friedman
STATUS
approved