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A216440 a(n) = smallest m such that 2n-1 | 2^m+1, or 0 if no such m exists. 1

%I #60 Apr 16 2023 08:39:14

%S 1,1,2,0,3,5,6,0,4,9,0,0,10,9,14,0,5,0,18,0,10,7,0,0,0,0,26,0,9,29,30,

%T 0,6,33,0,0,0,0,0,0,27,41,0,0,0,0,0,0,24,15,50,0,0,53,18,0,14,0,0,0,

%U 55,0,50,0,7,65,0,0,34,69,0,0,14,0,74,0,0,0,26,0

%N a(n) = smallest m such that 2n-1 | 2^m+1, or 0 if no such m exists.

%C From _Kristjan Kiolein_, Mar 25 2023: (Start)

%C Conjecture: a(n) is the number of riffle shuffles of 2(n-1) cards required to reverse the order of the deck, or 0 if no such number of shuffles exists.

%C The number of shuffles required to reverse the deck is A002326(n)/2 when a(n) != 0 and n != 1.

%C This conjecture is in the context of the in-shuffle variant of the riffle shuffle, e.g., [1, 2, 3, 4] -> [3, 1, 4, 2] -> [4, 3, 2, 1].

%C This does not apply in the context of the out-shuffle variant of the riffle shuffle, e.g., [1, 2, 3, 4] -> [1, 3, 2, 4] -> [1, 2, 3, 4]. (End)

%H V. Raman, <a href="/A216440/b216440.txt">Table of n, a(n) for n = 1..100</a>.

%o (PARI) for(i=0,200,i++;m=0;for(x=1,i,if(((2^x+1))%i==0,m=x;break));print1(m",")) \\ _V. Raman_, Nov 22 2012

%Y Cf. A002326.

%K nonn

%O 1,3

%A _V. Raman_, Sep 07 2012

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Last modified March 29 11:45 EDT 2024. Contains 371278 sequences. (Running on oeis4.)