OFFSET
0,1
COMMENTS
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.
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
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
LINKS
J. Brillhart et al., Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 2002.
EXAMPLE
p=333667 is the smallest prime such that multiplicative order of 10 modulo p is 3^2, so a(2)=333667.
PROG
(PARI) a(n) = factor(polcyclo(3^n, 10)/3)[1, 1] \\ Max Alekseyev, Nov 18 2014
CROSSREFS
KEYWORD
hard,more,nonn
AUTHOR
Farideh Firoozbakht, Oct 06 2006
EXTENSIONS
Revised and extended by Max Alekseyev, Apr 25 2009
a(13), a(15), a(17), a(20), a(21), a(24) from Ray Chandler, Dec 25 2013
STATUS
approved