A145342 - OEIS

%I #36 Apr 09 2019 05:10:08

%S 1,2,3,4,5,7,6,8,9,13,11,15,10,14,12,16,17,25,21,29,19,27,23,31,18,26,

%T 22,30,20,28,24,32,33,49,41,57,37,53,45,61,35,51,43,59,39,55,47,63,34,

%U 50,42,58,38,54,46,62,36,52,44,60,40,56,48,64,65,97,81,113,73,105,89

%N a(n) = (A145341(n) + 1)/2.

%C This sequence is a permutation of the positive integers. It is its own inverse permutation.

%C Fixed points of the permutation are the terms of A044051. - _Ivan Neretin_, Oct 31 2015

%C From _Yosu Yurramendi_, Feb 04 2019: (Start)

%C If the terms (n > 0) are written as an array (left-aligned fashion) with rows of length 2^m, m = 0,1,2,3,...

%C    1;

%C    2,  3;

%C    4,  5,  7,  6;

%C    8,  9, 13, 11, 15, 10, 14, 12;

%C   16, 17, 25, 21, 29, 19, 27, 23, 31, 18, 26, 22, 30, 20, 28, 24;

%C   32, 33, 49, 41, 57, 37, 53, 45, 61, 35, 51, 43, 59, 39, 55, 47, 63, 34, ...

%C then the following relationship can be observed:

%C a(1) = 1, a(2) = 2, a(3) = 3,

%C for m > 0, a(2^(m+1)) = 2*a(2^m), a(2^m + 1) = a(2^m) + 1, a(2^(m+1)+ 2^m) = 2*a(2^(m+1)) - 1, for 0 < k < 2^m, a(2^(m+1)+ k) = 2*a(2^m + k) - 1, a(2^(m+1)+ 2^m + k) = a(2^(m+1) + k) + 1

%C (End)

%H Ivan Neretin, <a href="/A145342/b145342.txt">Table of n, a(n) for n = 1..8192</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%t Table[(FromDigits[Reverse[IntegerDigits[2n-1, 2]], 2] +1)/2, {n, 71}] (* _Ivan Neretin_, Oct 31 2015 *)

%o (R)

%o nmax <- 10^3 # by choice

%o b <- vector()

%o for (o in seq(1,nmax,2)){

%o   w <- which(as.numeric(intToBits(o))==1)

%o   b <- c(b, sum(2^(max(w)-w)))

%o }

%o a <- (b+1)/2

%o a[1:71]

%o # _Yosu Yurramendi_, Feb 04 2019

%o (PARI) a(n) = (1+fromdigits(Vecrev(binary(2*n-1)), 2))/2; \\ _Michel Marcus_, Feb 04 2019

%Y Cf. A044051, A145341.

%K base,nonn

%O 1,2

%A _Leroy Quet_, Oct 08 2008

%E More terms from _R. J. Mathar_ and _Ray Chandler_, Oct 10 2008