A176351 Numbers n such that 2*3^n + 1 is a primitive prime factor of 10^3^n - 1.

4, 180, 320, 5480, 12096, 17720, 82780, 1175232

Offset: 1

Author

T. D. Noe, Apr 15 2010

Keywords

• hard
• more
• nonn

Comments

Consider the problem of finding the smallest number k such that the decimal representation of 1/k has period 3^e for a given e. The number k is usually 3^(e+2). However, if e is one of the n in this sequence, then the prime 2*3^n+1 is a smaller k. The first instance of these exceptions is 1/163, which has a period of 81.

Subsequence of A003306.

10 must be a square residue modulo 2*3^n + 1, implying that n must be a multiple of 4.

Crossrefs

Cf. A003306 (primes of the form 2*3^n+1), A003060 (least k such that 1/k has period n).

Programs

Mathematica

Select[Range[10000], PrimeQ[1+2*3^# ] && MultiplicativeOrder[10,1+2*3^# ] == 3^# &]

Extensions

Two more terms from Max Alekseyev, May 03 2010