A122787 a(n) is the smallest prime p such that the multiplicative order of 10 modulo p is 3^n.

3, 37, 333667, 757, 163, 411361786890737698932559, 313471, 2558791, 618846643, 2238862519, 396319276163359, 34720813

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

Table of n, a(n) for n=0..11.

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

Cf. A014950, A066364.

Sequence in context: A088098 A284411 A176245 * A241895 A093939 A129122

Adjacent sequences: A122784 A122785 A122786 * A122788 A122789 A122790

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