2,1

In other words, least k such that prime(n) divides (2^k-1), n>=2.

Concerning the complexity of computing this sequence, see for example Bach And Shallit, p. 115, exercise 8.

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

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

E. Bach and J. O. Shallit, Algorithmic Number Theory, I.

O. N. Karpenkov, On examples of difference operators ..., Funct. Anal. Other Math., 1 (2006), 175-180.

S. K. Sehgal, Group rings, pp. 455-541 in Handbook of Algebra, Vol. 3, Elsevier, 2003; see p. 493.

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.

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]

Nick Hobson and Zak Seidov, Table of n, a(n) for n = 2..5001

Gary W. Adamson, Further comments

a(n)=(A000040(n)-1)/A001917(n); a(A072190(n))=A001122(n)-1 - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 06 2004

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.

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 22 2009: (Start)

[Conway & Guy, p. 166]: referring to the work of Euler, 1/13 in base 2 =

.000100111011...; (cycle length of 12) (End)

with(numtheory): [ seq(order(2, ithprime(n)), n=2..60) ];

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]

(PARI) a(n)=if(n<0, 0, k=1; while((2^k-1)%prime(n)>0, k++); k)

(PARI) a(n)=if(n<0, 0, znorder(Mod(2, prime(n)))) - Nick Hobson (nickh(AT)qbyte.org), Jan 08 2007

Cf. A002326 (order of 2 mod 2n+1), A036116, A036117, A062117.

Cf. A065941 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 22 2009]

Sequence in context: A075363 A082382 A064691 this_sequence A093839 A096780 A143986

Adjacent sequences: A014661 A014662 A014663 this_sequence A014665 A014666 A014667

nonn

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 11 2003