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

Author

Farideh Firoozbakht, Oct 06 2006

Keywords

• hard
• more
• nonn

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

Examples

			p=333667 is the smallest prime such that multiplicative order of 10 modulo p is 3^2, so a(2)=333667.
		

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.

Crossrefs

Cf. A014950, A066364.

Programs

PARI

a(n) = factor(polcyclo(3^n,10)/3)[1,1] \\ Max Alekseyev, Nov 18 2014

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