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A137606 Numbers m such that all numbers {1...m} appear in the sequence {b(0) = m, b(n+1) = b(n)/2 if even, m-(b(n)+1)/2 otherwise}.

%I A137606 #15 Jun 08 2025 17:48:18
%S A137606 1,2,3,4,6,7,10,12,15,19,24,27,30,31,34,36,40,42,51,52,54,66,70,75,82,
%T A137606 84,87,90,91,96,99,100,106,114,120,132,135,136,147,156,159,174,175,
%U A137606 180,184,187,190,192,195,210,211,222,231,232,234,240,244,246,252,255,262
%N A137606 Numbers m such that all numbers {1...m} appear in the sequence {b(0) = m, b(n+1) = b(n)/2 if even, m-(b(n)+1)/2 otherwise}.
%C A137606 Lemma: A sequence {b(n)} defined as above with m>1 cannot have values outside [1,m]. (For m=1, b=(1,0,0,0....).)
%C A137606 Corollary: Such a sequence {b(n)} is periodic with period <= m (except maybe for some initial terms).
%C A137606 Lemma 2: For any m>1, b(1) = floor( m/2 ) and if b(n)=m-1, then b(n+1)= [ m/2 ].
%C A137606 Proposition: As soon as there is a term b(n)=2^k, the (b-)sequence continues b(n+1)=2^(k-1),...,b(n+k)=1, b(n+k+1)=m-1 and then starts over with b(n+k+2)=b(1).
%C A137606 Corollary 2: Numbers m=2^k, k>2 cannot appear in the present sequence.
%C A137606 Proposition: For any b(0)=m>1, sooner or later the value 1 is reached.
%C A137606 Generate a sequence b(n) by the following rule. If b(n-1) is divisible by 2 then b(n) = b(n-1)/2. If b(n-1) is not divisible by 2 then b(n) = b(0)-(b(n-1)+1)/2. When b(n)=1 it ends. Sequence gives all m such that all numbers k with 1<=k<=m-2 appear in b(n), b(0)=m.
%C A137606 Sequence contains 1 and numbers m>1 such that 2m-1 is prime and -2 or 2 is a primitive root modulo 2m-1. - _Max Alekseyev_, May 16 2008
%H A137606 Robert G. Wilson v, <a href="/A137606/b137606.txt">Table of n, a(n) for n = 1..2354</a>.
%F A137606 This sequence as a set is the union of { 1 }, { (A105874(n)+1)/2 } and { (A001122(n)+1)/2 }. - _Max Alekseyev_, May 16 2008
%e A137606 6->3->4->2->1. 1,2,3,4=6-2 appear in b(n), b(0)=6. So 6 is a term of A137606.
%t A137606 f[n_] := Block[{lst = {n}, a}, While[a = Last@ lst; a != 1, AppendTo[lst, If[ EvenQ@ a, a/2, lst[[1]] - (a + 1)/2]]]; Length@ lst - 1]; t = Array[f, 262]; Select[ Range @ 262, t[[ # ]] == # - 2 &] (* _Robert G. Wilson v_ *)
%o A137606 (PARI) b137606(n)= n=[n]; for( i=1,n[1]-1, n=concat( n, if( n[i]%2, n[1]-(n[i]+1)/2, n[i]/2 )); n[i]>1 || break); n
%o A137606 A137606(Nmax) = for( n=1,Nmax, n==#b137606(n) && print1(n","))
%o A137606 (PARI) forprime(p=3,10^3, if(znorder(Mod(-2,p))==p-1||znorder(Mod(2,p))==p-1, print1((p+1)/2,", ") )) \\ _Max Alekseyev_, May 16 2008
%Y A137606 Cf. A137607, A096259, A105874, A001122.
%K A137606 nonn
%O A137606 1,2
%A A137606 _Yasutoshi Kohmoto_, Apr 23 2008
%E A137606 Edited & extended by _M. F. Hasler_, Apr 28 2008