A054471 Smallest prime p having n different cycles in decimal expansions of k/p, k=1..p-1.

7, 3, 103, 53, 11, 79, 211, 41, 73, 281, 353, 37, 2393, 449, 3061, 1889, 137, 2467, 16189, 641, 3109, 4973, 11087, 1321, 101, 7151, 7669, 757, 38629, 1231, 49663, 12289, 859, 239, 27581, 9613, 18131, 13757, 33931, 9161, 118901, 6763, 18233

Offset: 1

Author

Robert G. Wilson v, 1994; Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 22 2000

Keywords

• nonn
• easy
• nice
• base

Comments

First cyclic number of n-th degree (or n-th order): the reciprocals of these numbers belong to one of n different cycles. Each cycle has (a(n) - 1)/n digits.

From Robert G. Wilson v, Aug 21 2014: (Start)

recursive by indices:

1, 7, 211, 79337, 634776923741, ...

2, 3, 103, 2368589, 785245568161181, ...

4, 53, 135257, 2332901103899, ...

5, 11, 353, 3795457, 693814982285339, ...

6, 79, 26861, 23947548497, ...

8, 41, 118901, 1015118238709, ...

9, 73, 142789, 267291583927, ...

10, 281, 3097183, 66880786504811, ...

12, 37, 18131, 105385168331, ...

13, 2393, 11160953, 7140939250711817, ...

14, 4999, 2148340247, > 10^19,

... .

(End)

References

• John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 162.
• M. Gardner, Mathematical Circus, Cambridge University Press (1996).

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..65000 (first 1000 terms from _T. D. Noe_)
• H. Richter, The period length of the decimal expansion of a fraction
• Index entries for sequences related to decimal expansion of 1/n

Crossrefs

First time n appears in A006556.

Cf. A006883, A097443, A055628, A056157, A056210, A056211, A056212, A056213, A056214, A056215, A056216, A056217, A098680, which are sequences of primes p where the period of the reciprocal is (p-1)/n for n=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13.

Cf. A101208, A101209 (similar sequences for base 2 and base 3).

Programs

Mathematica

a[n_Integer] := Block[{m = If[ OddQ@ n, 2n, n]}, p = m +1; While[ !PrimeQ@ p || p != 1 + n*MultiplicativeOrder[10, p], p = p += m]; p]; a[1] = 7; a[4] = 53; Array[f, 50] (* Robert G. Wilson v, Apr 19 2005; revised Aug 20 2014 and Feb 14 2025 *)

Extensions

More terms from David W. Wilson, May 22 2000