This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Starting from the second most significant bit, we continue complementing every second bit (in this case, starting from the second most significant bit), as long as the first zero is encountered, which is also complemented if its distance to the most significant bit is odd, after which the remaining bits are left intact. E.g. 121 = 1111001 in binary. Complementing its second and fourth most significant bits (positions 5 & 3) and stopping at the first zero-bit at position 2 (which is not complemented, as its distance to the msb is 6), we obtain "10100.." after which the rest of the bits stay same, so we get 1010001, which is 81's binary representation, thus a(121)=81. On the other hand, 125 = 1111101 in binary and the transducer complements the bits at positions 5, 3 and also the first zero at the position 1 (because at odd distance from the msb), yielding 101011., after which the remaining bit stays same, thus we get 1010111, which is 87's binary representation, thus a(125)=87.
a := n -> if n < 2 then n elif convert(convert(n+1, base, 2), `+`) = 1 then (n+1)/2 else n+1 fi: seq(a(n), n=0..71); # Peter Luschny, Jul 16 2016
Table[If[IntegerQ@ Log2[n + 1], (n + 1)/2, n + 1], {n, 0, 71}] /. Rational -> 0 (* _Michael De Vlieger, Jul 13 2016 *)
def ok(n): return n&(n - 1)==0 def a(n): return n if n<2 else (n + 1)/2 if ok(n + 1) else n + 1 # Indranil Ghosh, Jun 09 2017
maxlevel <- 5 # by choice
a <- 1
for(m in 1:maxlevel){
a[2^m ] <- 2^m + 1
a[2^(m+1) - 1] <- 2^m
for (k in 0:(2^m-2)){
a[2^(m+1) + 2*k + 1] <- 2*a[2^m + k]
a[2^(m+1) + 2*k + 2] <- 2*a[2^m + k] + 1}
}
a <- c(0, a)
# Yosu Yurramendi, Sep 05 2020
25 = 11001 in binary, the first zero-bit at odd distance from the msb is at position 1 (distance 3) and the first one-bit at even distance from the msb is at position 0 (distance 4), thus we stop at the former, after complementing the bits 3-1, which gives us 10111 (23 in binary), thus a(25)=23.
maxlevel <- 5 # by choice
a <- 1
for(m in 0:maxlevel) {
for(k in 0:(2^m-1)) {
a[2^(m+1) + 2*k ] <- 2*a[2^m + k]
a[2^(m+1) + 2*k + 1] <- 2*a[2^m + k] + 1
}
x <- floor(2^m*5/3)
a[2*x ] <- 2*a[x] + 1
a[2*x + 1] <- 2*a[x]
}
(a <- c(0, a))
# Yosu Yurramendi, Oct 12 2020
A316385(42) = 50 hence a(50) = 42.
b1(n) = my(b=binary(n)); fromdigits(concat(b[1], Vecrev(vector(#b-1, k, b[k+1]))), 2); \\ A059893 b2(n) = if(n < 2, n, if((n + 1) == 2^logint(n + 1, 2), (n + 1) / 2, n + 1)) \\ A153152 a(n) = my(A = 2^logint(n, 2), B = b1(b2(b1(n))) - A); (2 * B + 1) * A / 2 ^ (if(B == 0, -1, logint(B, 2)) + 1) \\ Mikhail Kurkov, Sep 09 2023 [verification needed]
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