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%I #22 Nov 18 2014 09:04:10
%S 3,37,333667,757,163,411361786890737698932559,313471,2558791,
%T 618846643,2238862519,396319276163359,34720813
%N a(n) is the smallest prime p such that the multiplicative order of 10 modulo p is 3^n.
%C For n>0, a(n) is the smallest prime p>3 such that 3^n*p but not 3^(n-1)*p is a solution to 10^x==1 (mod x). A014950 gives solutions of this equation. It's obvious that if n is a term of A014950 then 3n is also a term of A014950. So according to the definition of a(n), for each m>n-1, 3^m*a(n) is in the sequence A014950.
%C a(n) is the smallest prime divisor of \Phi_{3^n}(10)/3, where \Phi_k(x) is k-th cyclotomic polynomial. a(n) is congruent to 1 modulo 3^n and 1, 3, 9, 13, 27, 31, 37, or 39 modulo 40. - _Max Alekseyev_, Nov 18 2014
%C a(12)>10^17, a(13)=796884087799, a(14)=86093443, a(15)=70367039929, a(16)>8*10^18, a(17)=662489036191, a(18)>10^19, a(19)>10^19, a(20)=38180289190951, a(21)=28305715767319, a(22)>10^20, a(23)>10^20, a(24)=63829075244707. - _Ray Chandler_, Dec 25 2013
%H J. Brillhart et al., <a href="http://dx.doi.org/10.1090/conm/022">Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers</a>, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 2002.
%e p=333667 is the smallest prime such that multiplicative order of 10 modulo p is 3^2, so a(2)=333667.
%o (PARI) a(n) = factor(polcyclo(3^n,10)/3)[1,1] \\ _Max Alekseyev_, Nov 18 2014
%Y Cf. A014950, A066364.
%K hard,more,nonn
%O 0,1
%A _Farideh Firoozbakht_, Oct 06 2006
%E Revised and extended by _Max Alekseyev_, Apr 25 2009
%E a(13), a(15), a(17), a(20), a(21), a(24) from _Ray Chandler_, Dec 25 2013
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