Search: id:A014664
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%I A014664
%S A014664 2,4,3,10,12,8,18,11,28,5,36,20,14,23,52,58,60,66,35,9,39,82,11,48,100,
%T A014664 51,106,36,28,7,130,68,138,148,15,52,162,83,172,178,180,95,96,196,99,
%U A014664 210,37,226,76,29,119,24,50,16,131,268,135,92,70,94,292,102,155,156,316
%N A014664 Order of 2 mod n-th prime.
%C A014664 In other words, least k such that prime(n) divides (2^k-1), n>=2.
%C A014664 Concerning the complexity of computing this sequence, see for example 
               Bach And Shallit, p. 115, exercise 8.
%C A014664 Also A002326((p_n-1)/2). Conjecture. If p_n is not a Wieferich prime 
               (1093, 3511,...) then A002326(((p_n)^k-1)/2)=a(n)*(p_n)^(k-1) - Vladimir 
               Shevelev (shevelev(AT)bgu.ac.il), May 26 2008
%C A014664 If for distinct i,j,...,k we have a(i)=a(j)=...=a(k) then the number 
               N=p_i*p_j*...*p_k is in A001262 and moreover A137576((N-1)/2)=N. 
               For example, a(16)=a(37)=a(255)=52. Therefore we could take N=p_16*p_37*p_255=53*157*1613=13421773. 
               - Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jun 14 2008
%D A014664 E. Bach and J. O. Shallit, Algorithmic Number Theory, I.
%D A014664 O. N. Karpenkov, On examples of difference operators ..., Funct. Anal. 
               Other Math., 1 (2006), 175-180.
%D A014664 S. K. Sehgal, Group rings, pp. 455-541 in Handbook of Algebra, Vol. 3, 
               Elsevier, 2003; see p. 493.
%D A014664 Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1966; 
               Table 48, page 98, "Exponents to Which a Belongs, MOD p and MOD p^n.
%D A014664 John H. Conway and Richard Guy, "The Book of Numbers", Springer-Verlag, 
               1996; p. 166: "How does the Cycle Length Change with the Base?". 
               [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 22 2009]
%H A014664 Nick Hobson and Zak Seidov, <a href="b014664.txt">Table of n, a(n) for 
               n = 2..5001</a>
%H A014664 Gary W. Adamson, <a href="a014664.txt">Further comments</a>
%F A014664 a(n)=(A000040(n)-1)/A001917(n); a(A072190(n))=A001122(n)-1 - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Jun 06 2004
%e A014664 2^2 == 1 mod 3 and so a(2) = 2; 2^4 == 1 mod 5 and so a(3) = 4; 2^3 == 
               1 mod 7 and so a(4) = 3; 2^10 == 1 mod 11 and so a(5) = 10; etc.
%e A014664 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 22 2009: 
               (Start)
%e A014664 [Conway & Guy, p. 166]: referring to the work of Euler, 1/13 in base 
               2 =
%e A014664 .000100111011...; (cycle length of 12) (End)
%p A014664 with(numtheory): [ seq(order(2,ithprime(n)), n=2..60) ];
%t A014664 Reap[Do[p=Prime[i];Do[If[PowerMod[2,k,p]==1,Print[{i,k}];Sow[{i,k}];Goto[ni]],
               {k,1,10^6}];Label[ni],{i,2,5001}]][[2,1]] [From Zak Seidov (zakseidov(AT)yahoo.com), 
               Jan 26 2009]
%o A014664 (PARI) a(n)=if(n<0,0,k=1;while((2^k-1)%prime(n)>0,k++);k)
%o A014664 (PARI) a(n)=if(n<0, 0, znorder(Mod(2, prime(n)))) - Nick Hobson (nickh(AT)qbyte.org), 
               Jan 08 2007
%Y A014664 Cf. A002326 (order of 2 mod 2n+1), A036116, A036117, A062117.
%Y A014664 Cf. A065941 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 22 2009]
%Y A014664 Sequence in context: A075363 A082382 A064691 this_sequence A093839 A096780 
               A143986
%Y A014664 Adjacent sequences: A014661 A014662 A014663 this_sequence A014665 A014666 
               A014667
%K A014664 nonn
%O A014664 2,1
%A A014664 N. J. A. Sloane (njas(AT)research.att.com).
%E A014664 More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 11 2003

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