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The odd integers in the sequence appear in blocks and the successive sizes of these blocks are given by the sequence itself.
1

%I #45 Aug 03 2016 09:04:17

%S 1,2,3,5,4,6,7,9,11,8,10,12,13,15,17,19,21,14,16,18,20,22,23,25,27,29,

%T 24,26,28,30,31,33,35,37,39,41,32,34,36,38,40,42,43,45,47,49,51,53,55,

%U 44,46,48,50,52,54,56,57,59,61,63,65,67,69,71,73,58,60,62,64,66,68,70,72,74,75,77,79,81,83,85,87,89,91,93,95,76,78,80,82,84,86,88,90

%N The odd integers in the sequence appear in blocks and the successive sizes of these blocks are given by the sequence itself.

%C The sequence starts with a(1)=1 and is always extended with the smallest integer not yet in the sequence which does not lead to a contradiction. The sequence is a permutation of the natural numbers.

%H Zak Seidov, <a href="/A274500/b274500.txt">Table of n, a(n) for n = 1..10000</a>

%H Zak Seidov, <a href="/A274500/a274500.png">Color graph for n = 1..150.</a>

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

%e The blocks of odd integers are indicated here by parentheses; the successive block-sizes are1,2,3,5,4,6,7,9,11,8,10,... which reproduces the sequence itself:

%e (1),2,(3,5),4,6,(7,9,11),8,10,12,(13,15,17,19,21),14,16,18,20,22,(23,25,27,29),24,26,28,30,(31,33,35,37,39,41),

%e 32,34,36,38,40,42,(43,45,47,49,51,53,55),44,46,48,50,52,54,56,(57,59,61,63,65,67,69,71,73),58,60,62,64,66,68,70,72,74,

%e (75,77,79,81,83,85,87,89,91,93,95),76,78,80,82,84,86,88,90,92,94,96,(97,99,101,103,105,107,109,111),

%e 98,100,102,104,106,108,110,112,(113,115,117,119,121,123,125,127,129,131),...

%t s={1,2,3,5};k=2;od={3,5};Do[ev=Range[od[[1]]+1,od[[-1]]+1,2];s=Flatten@{s,ev};a=ev[[-1]]+1;k++;od=Range[a,a+2*s[[k]]-2,2];s=Flatten@{s,od},{100}];s

%Y Cf. A275119, A275335.

%K nonn,base

%O 1,2

%A _Zak Seidov_ and _Eric Angelini_, Jul 28 2016